Displaying 1-10 of 18 results found.
E.g.f.: exp(9*x*G(x)^8) / G(x)^8 where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.
+20
11
1, 1, 9, 225, 10017, 656289, 57255849, 6262226721, 825067217025, 127305462542913, 22527254639457801, 4498536675388410081, 1000890043482114644769, 245556248365681036646625, 65862976584851401437170217, 19174678419336874098038167329, 6022064808176665662053835550209
FORMULA
Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies: (1) A'(x)/A(x) = G(x)^8.
(2) A'(x) = exp(8*x*G(x)^8).
(3) A(x) = exp( Integral G(x)^8 dx ).
(4) A(x) = exp( Sum_{n>=1} A234513(n-1)*x^n/n ), where A234513(n-1) = binomial(9*n-2,n)/(8*n-1). (5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251589. (6) A(x) = Sum_{n>=0} A251589(n)*(x/A(x))^n/n! and (7) [x^n/n!] A(x)^(n+1) = (n+1)* A251589(n), where A251589(n) = 9^(n-7) * (n+1)^(n-9) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969). a(n) = Sum_{k=0..n} 9^k * n!/k! * binomial(9*n-k-9, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 128*(2*n-3)*(4*n-7)*(4*n-5)*(8*n-15)*(8*n-13)*(8*n-11)*(8*n-9)*(59049*n^7 - 1102248*n^6 + 8858079*n^5 - 39764115*n^4 + 107806473*n^3 - 176772075*n^2 + 162618742*n - 64907105)*a(n) = 81*(282429536481*n^15 - 8943601988565*n^14 + 132044525265870*n^13 - 1206188364304287*n^12 + 7627178203628841*n^11 - 35382975568258428*n^10 + 124478964551078775*n^9 - 338415281830783431*n^8 + 717436315214480025*n^7 - 1187215577095780764*n^6 + 1522794566607803919*n^5 - 1488866286016780047*n^4 + 1075889068341959448*n^3 - 543536112365518695*n^2 + 172059320987344825*n - 25799292366848000)*a(n-1) - 387420489*(59049*n^7 - 688905*n^6 + 3484620*n^5 - 9940725*n^4 + 17352558*n^3 - 18650247*n^2 + 11527801*n - 3203200)*a(n-2). - Vaclav Kotesovec, Dec 07 2014 a(n) ~ 9^(9*(n-1)-1/2) / 8^(8*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 9*x^2/2! + 225*x^3/3! + 10017*x^4/4! + 656289*x^5/5! +...
such that A(x) = exp(9*x*G(x)^8) / G(x)^8
where G(x) = 1 + x*G(x)^9 is the g.f. of A062994: G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
Note that
A'(x) = exp(9*x*G(x)^8) = 1 + 9*x + 225*x^2/2! + 10017*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 8*x^2/2 + 200*x^3/3 + 8976*x^4/4 + 592368*x^5/5 +...
and so A'(x)/A(x) = G(x)^8.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1, 9, 225, 10017, 656289, 57255849, 6262226721, ...];
n=2: [1, 2, 20, 504, 22320, 1453248, 126104256, 13731880320, ...];
n=3: [1, 3, 33, 843, 37233, 2411667, 208241361, 22581193851, ...];
n=4: [1, 4, 48, 1248, 55104, 3554496, 305558784, 33002857728, ...];
n=5: [1, 5, 65, 1725, 76305, 4906965, 420159825, 45211985325, ...];
n=6: [1, 6, 84, 2280, 101232, 6496704, 554376384, 59448214656, ...];
n=7: [1, 7, 105, 2919, 130305, 8353863, 710786601, 75977951175, ...];
n=8: [1, 8, 128, 3648, 163968, 10511232, 892233216, 95096756736, ...]; ...
in which the main diagonal begins (see A251587): [1, 2, 33, 1248, 76305, 6496704, 710786601, 95096756736, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 9^(n-7) * (n+1)^(n-8) * (262144*n^7 + 2494464*n^6 + 10470208*n^5 + 25229505*n^4 + 37857568*n^3 + 35537670*n^2 + 19414368*n + 4782969) for n>=0.
MATHEMATICA
Flatten[{1, 1, Table[Sum[9^k * n!/k! * Binomial[9*n-k-9, n-k] * (k-1)/(n-1), {k, 0, n}], {n, 2, 20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)
PROG
(PARI) {a(n) = local(G=1); for(i=1, n, G=1+x*G^9 +x*O(x^n)); n!*polcoeff(exp(9*x*G^8)/G^8, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if(n==0|n==1, 1, sum(k=0, n, 9^k * n!/k! * binomial(9*n-k-9, n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))
E.g.f.: exp(9*x*G(x)^8) / G(x) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994.
+20
10
1, 8, 191, 8310, 537117, 46444164, 5047987707, 662002733394, 101779688986425, 17959176833948928, 3578033935192224951, 794559576204365478318, 194620831940208238831701, 52129134740350115227721340, 15158273263608217360939225587, 4755712518628181890216523759754
FORMULA
Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then the e.g.f. A(x) of this sequence satisfies: (1) A'(x)/A(x) = G(x)^8 + 7*G'(x)/G(x).
(2) A(x) = F(x/A(x)^8) where F(x) is the e.g.f. of A251699. (3) A(x) = Sum_{n>=0} A251699(n)*(x/A(x)^8)^n/n! where A251699(n) = (7*n+1) * (8*n+1)^(n-2) * 9^n. (4) [x^n/n!] A(x)^(8*n+1) = (7*n+1) * (8*n+1)^(n-1) * 9^n.
a(n) = Sum_{k=0..n} 9^k * n!/k! * binomial(9*n-k-2,n-k) * (8*k-1)/(8*n-1) for n>=0.
Recurrence: 128*(2*n-1)*(4*n-3)*(4*n-1)*(8*n-7)*(8*n-5)*(8*n-3)*(8*n-1)*(33480783*n^8 - 453319173*n^7 + 2697889761*n^6 - 9230277240*n^5 + 19886167926*n^4 - 27672715746*n^3 + 24328423881*n^2 - 12365760717*n + 2776106045)*a(n) = 81*(160137547184727*n^16 - 2968899287488272*n^15 + 25604779347830979*n^14 - 136506824772659775*n^13 + 504285657127489314*n^12 - 1371500076773316825*n^11 + 2847804013092225933*n^10 - 4619534029925962572*n^9 + 5937710241656343834*n^8 - 6090889132598477481*n^7 + 4986522977501530773*n^6 - 3228624422259256476*n^5 + 1615386846720554091*n^4 - 595058403096826425*n^3 + 145565831993332122*n^2 - 17972427186502245*n - 2554359808000)*a(n-1) - 387420489*(33480783*n^8 - 185472909*n^7 + 462117474*n^6 - 687717459*n^5 + 680611896*n^4 - 464268429*n^3 + 210617505*n^2 - 51824070*n - 4480)*a(n-2). - Vaclav Kotesovec, Dec 07 2014 a(n) ~ 7 * 3^(18*n-3) / 8^(8*n-1/2) * n^(n-1) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014
EXAMPLE
E.g.f.: A(x) = 1 + 8*x + 191*x^2/2! + 8310*x^3/3! + 537117*x^4/4! + 46444164*x^5/5! +...
such that A(x) = exp(9*x*G(x)^8) / G(x)
where G(x) = 1 + x*G(x)^9 is the g.f. of A062994: G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
MATHEMATICA
Table[Sum[9^k * n!/k! * Binomial[9*n-k-2, n-k] * (8*k-1)/(8*n-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Dec 07 2014 *)
PROG
(PARI) {a(n)=local(G=1); for(i=0, n, G = 1 + x*G^9 +x*O(x^n)); n!*polcoeff(exp(9*x*G^8)/G, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, 9^k * n!/k! * binomial(9*n-k-2, n-k) * (8*k-1)/(8*n-1) )}
for(n=0, 20, print1(a(n), ", "))
Number of dissections of a polygon: binomial(4*n, n)/(3*n + 1).
(Formerly M3587 N1454)
+10
210
1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 225568798, 1882933364, 15875338990, 134993766600, 1156393243320, 9969937491420, 86445222719724, 753310723010608, 6594154339031800, 57956002331347120, 511238042454541545
COMMENTS
The number of rooted loopless n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005 Number of lattice paths from (1,0) to (3*n+1,n) which, starting from (1,0), only utilize the steps +(1,0) and +(0,1) and additionally, the paths lie completely below the line y = (1/3)*x (i.e., if (a,b) is in the path, then b < a/3). - Joseph Cooper (jecooper(AT)mit.edu), Feb 07 2006
Number of length-n restricted growth strings (RGS) [s(0), s(1), ..., s(n-1)] where s(0) = 0 and s(k) <= s(k-1) + 3, see fxtbook link below. - Joerg Arndt, Apr 08 2011 a(n), n >= 1, enumerates quartic trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m = 4. See the Graham et al. reference, p. 347. eq. 7.66. (Second edition, p. 361, eq. 7.67.) See also the Pólya-Szegő reference.
Also 4-Raney sequence. See the Graham et al. reference, pp. 346-347.
(End)
Bacher: "We describe the statistics of checkerboard triangulations obtained by coloring black every other triangle in triangulations of convex polygons." The current sequence ( A002293) occurs on p. 12 as one of two "extremal sequences" of an array of coefficients of polynomials, whose generating functions are given in terms of hypergeometric functions. - Jonathan Vos Post, Oct 05 2007 A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. With D(z,t) that g.f., a g.f. for signed A002293 is {[-1+1/D(z,t)]/(4t)}^(1/3). - Tom Copeland, Oct 10 2012 For relations to compositional inversion, the Legendre transform, and convex geometry, see the Copeland, the Schuetz and Whieldon, and the Gross (p. 58) links. - Tom Copeland, Feb 21 2017 (See also Gross et al. in A062994. - Tom Copeland, Dec 24 2019) This is the number of A'Campo bicolored forests of degree n and co-dimension 0. This can be shown using generating functions or a combinatorial approach. See Combe and Jugé link below. - Noemie Combe, Feb 28 2017 Conjecturally, a(n) is the number of 3-uniform words over the alphabet [n] that avoid the patterns 231 and 221 (see the Defant and Kravitz link). - Colin Defant, Sep 26 2018 The compositional inverse o.g.f. pair in Copeland's comment above are related to a pair of quantum fields in Balduf's thesis by Theorem 4.2 on p. 92. Cf. A001764. - Tom Copeland, Dec 13 2019 a(n) is the total number of down steps before the first up step in all 3_1-Dyck paths of length 4*n. A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. - Sarah Selkirk, May 10 2020 a(n) is the number of pairs (A<=B) of noncrossing partitions of [2n] such that every block of A has exactly two elements. In fact, it is proved that a(n) is the number of planar tied arc diagrams with n arcs (see Aicardi link below). A planar diagram with n arcs represents a noncrossing partition A of [2n] with n blocks, each block containing the endpoints of one arc; each tie connects two arcs, so that the ties define a partition B >= A: the endpoints of two arcs connected by a tie belong to the same block of B. Ties do not cross arcs nor other ties iff B has a planar diagram, i.e., B is a noncrossing partition. - Francesca Aicardi, Nov 07 2022 Dropping the initial 1 (starting 1, 4, 22 with offset 1) yields the REVERT transformation 1, -4 ,10, -20, 35.. essentially A000292 without leading 0. - R. J. Mathar, Aug 17 2023 Number of rooted polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024 This is instance k = 4 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - Wolfdieter Lang, Feb 05 2024
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
Peter Hilton and Jean Pedersen, Catalan numbers, their generalization, and their uses, Math. Intelligencer 13 (1991), no. 2, 64-75.
V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures] D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (T_n for s=4). C. O. Oakley and R. J. Wisner, Flexagons, Am. Math. Monthly 64 (3) (1957) 143-154, u_{3k+1}.
FORMULA
O.g.f. satisfies: A(x) = 1 + x*A(x)^4 = 1/(1 - x*A(x)^3).
a(n) = binomial(4*n,n-1)/n, n >= 1, a(0) = 1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
Integral representation as n-th Hausdorff power moment of a positive function on the interval [0, 256/27]:
a(n) = Integral_{x=0..256/27}(x^n((3/256) * sqrt(2) * sqrt(3) * ((2/27) * 3^(3/4) * 27^(1/4) * 256^(/4) * hypergeom([-1/12, 1/4, 7/12], [1/2, 3/4], (27/256)*x)/(sqrt(Pi) * x^(3/4)) - (2/27) * sqrt(2) * sqrt(27) * sqrt(256) * hypergeom([1/6, 1/2, 5/6], [3/4, 5/4], (27/256)*x)/ (sqrt(Pi) * sqrt(x)) - (1/81) * 3^(1/4) * 27^(3/4) * 256^(1/4) * hypergeom([5/12, 3/4, 13/12], [5/4, 3/2], (27/256)*x/(sqrt(Pi)*x^(1/4)))/sqrt(Pi))).
This representation is unique as it represents the solution of the Hausdorff moment problem.
O.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 4/3], (256/27)*x);
E.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 1, 4/3], (256/27)*x). (End)
a(n) = upper left term in M^n, M = the production matrix:
1, 1
3, 3, 1
6, 6, 3, 1
...
(where 1, 3, 6, 10, ...) is the triangular series. - Gary W. Adamson, Jul 08 2011 O.g.f. satisfies g = 1+x*g^4. If h is the series reversion of x*g, so h(x*g)=x, then (x-h(x))/x^2 is the o.g.f. of A006013. - Mark van Hoeij, Nov 10 2011 a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1-i} Sum_{k=0..n-1-i-j} a(i)*a(j)*a(k)*a(n-1-i-j-k) for n>=1; and a(0) = 1. - Robert FERREOL, Apr 02 2015 a(n) ~ 2^(8*n+1/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n+3/2)). - Vaclav Kotesovec, Jun 03 2015 From Peter Bala, Oct 16 2015: (Start)
D-finite with recurrence: a(n+1) = a(n)*4*(4*n + 3)*(4*n + 2)*(4*n + 1)/((3*n + 2)*(3*n + 3)*(3*n + 4)). - Chai Wah Wu, Feb 19 2016 E.g.f.: F([1/4, 1/2, 3/4], [2/3, 1, 4/3], 256*x/27), where F is the generalized hypergeometric function. - Stefano Spezia, Dec 27 2019 a(n) = hypergeom([1 - n, -3*n], [2], 1). Row sums of A173020. - Peter Bala, Aug 31 2023 G.f.: t*exp(4*t*hypergeom([1, 1, 5/4, 3/2, 7/4], [4/3, 5/3, 2, 2], (256*t)/27))+1. - Karol A. Penson, Dec 20 2023 a(n) = Integral_{x=0..256/27} x^(n)*W(x)dx, n>=0, where W(x) = x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi), with R(x) = ((i + sqrt(3))*(4*sqrt(256 - 27*x) -12*i*sqrt(3*x))^(1/3))/16 - ((i - sqrt(3))*(4*sqrt(256 - 27*x) + 12*i* sqrt(3*x))^(1/3))/16, where i is the imaginary unit.
The elementary function W(x) is positive on the interval x = (0, 256/27) and is equal to the combination of hypergeometric functions in my formula from 2010; see above.
(Pi*W(x))^6 satisfies an algebraic equation of order 6, with integer polynomials as coefficients. (End)
EXAMPLE
There are a(2) = 4 quartic trees (vertex degree <= 4 and 4 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these four trees yields 4*4 + 6 = 22 = a(3) such trees.
MAPLE
series(RootOf(g = 1+x*g^4, g), x=0, 20); # Mark van Hoeij, Nov 10 2011 # Using the integral representation above:
Digits:=6;
R:=proc(x)((I + sqrt(3))*(4*sqrt(256 - 27*x) - 12*I*sqrt(3)*sqrt(x))^(1/3))/16 - ((I - sqrt(3))*(4*sqrt(256 - 27*x) + 12*I*sqrt(3)*sqrt(x))^(1/3))/16; end;
W:=proc(x) x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi); end;
# Attention: W(x) is singular at x = 0. Integration is done from a very small positive x to x = 256/27.
# For a(8): ... gives 420732
evalf(int(x^8*W(x), x=0.000001..256/27));
MATHEMATICA
CoefficientList[InverseSeries[ Series[ y - y^4, {y, 0, 60}], x], x][[Range[2, 60, 3]]]
Table[Binomial[4n, n]/(3n+1), {n, 0, 25}] (* Harvey P. Dale, Apr 18 2011 *) CoefficientList[1 + InverseSeries[Series[x/(1 + x)^4, {x, 0, 60}]], x] (* Gheorghe Coserea, Aug 12 2015 *) terms = 22; A[_] = 0; Do[A[x_] = 1 + x*A[x]^4 + O[x]^terms, terms];
PROG
(PARI) my(x='x+O('x^33)); Vec(1 + serreverse(x/(1+x)^4)) \\ Gheorghe Coserea, Aug 12 2015 (Python)
for n in range(100):
x = x*4*(4*n+3)*(4*n+2)*(4*n+1)//((3*n+2)*(3*n+3)*(3*n+4))
(GAP) List([0..22], n->Binomial(4*n, n)/(3*n+1)); # Muniru A Asiru, Nov 01 2018
CROSSREFS
Cf. A006632, A006633, A006634, A025174, A069271, A196678, A224274, A233658, A233666, A233667, A277877, A283049, A283101, A283102, A283103. Cf. A130564 (for generalized Catalan C(k, n), for = 4).
Member k=5 of a family of generalized Catalan numbers.
+10
27
1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 180320238280, 2386316821325, 31864803599700, 428798445360120, 5809228810425801, 79168272296871450, 1084567603590147950
COMMENTS
The generalized Catalan numbers C(k,n):= binomial(k*n+1,n)/(k*n+1) become for negative k=-|k|, with |k|>=2, ((-1)^(n-1))*binomial((|k|+1)*n-2,n)/(|k|*n-1), n>=0.
The family c(k,n):=binomial((k+1)*n-2,n)/(k*n-1), n>=1, has the members A000108, A006013, A006632, A118971 for k=1,2,3,4, respectively (but the offset there is 0). The members of the C(k,n) family for positive k are: A000012 (powers of 1), A000108, A001764, A002293, A002294, A002295, A002296, A007556, A062994, for k=1..9. The ordinary generating functions for the k-family {c(k, n+1)}_{n>=0}, k >= 1, are G(k, x) = hypergeometric(Aseq(k+1), Bseq(k), ((k+1)^(k+1)/k^k)*x), with Aseq(k+1) = [a(k)_1,..., a(k)_{k+1}], where a(k)_j = (2*k - (j-1))/(k+1), and Bseq(k) = [b(k)_1, ..., b(k)_k], where b(k)_j = (2*k - (j-1))/k. The e.g.f. has an extra 1 in the B-section, which leads to a cancellation with the A-section 1 term. Thanks to Dixon J. Jones for asking for the general formulas. - Wolfdieter Lang, Feb 04 2024 The o.g.f. of {C(k, n)}_{n>=0} is in the Graham-Knuth-Patashnik book denoted as B_k(z) on pp. 200, 349 (2nd ed. 1994, pp. 200, 363). - Wolfdieter Lang, Mar 07 2024
REFERENCES
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1994, pp. 200, 363.
LINKS
K. Kobayashi, H. Morita and M. Hoshi, Coding of ordered trees, Proceedings, IEEE International Symposium on Information Theory, ISIT 2000, Sorrento, Italy, Jun 25 2000.
FORMULA
a(n) = binomial((k+1)*n-2,n)/(k*n-1), with k=5.
G.f.: inverse series of y*(1-y)^5.
G.f.: (5/6)*(1 - hypergeom([-1, 1, 2, 3, 4]/6, [1, 2, 3, 4]/5,(6^6/5^5)*x)).
E.g.f.: (5/6)*(1 - hypergeom([-1, 1, 2, 3, 4]/6, [1, 2, 3, 4, 5]/5,(6^6/5^5)*x)). (End)
D-finite with recurrence 5*n*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n) -72*(6*n-7)*(3*n-1)*(2*n-1)*(3*n-2)*(6*n-5)*a(n-1)=0. - R. J. Mathar, May 07 2021
MATHEMATICA
Rest@ CoefficientList[InverseSeries[Series[y (1 - y)^5, {y, 0, 18}], x], x] (* Michael De Vlieger, Oct 13 2019 *)
CROSSREFS
Cf. A000012, A000108, A001764, A002293, A002294, A002295, A002296, A006013, A062994, A006632, A007556, A118971, A130565, A234466, A234513, A234573, A235340.
1, 1, 10, 145, 2470, 46060, 910252, 18730855, 397089550, 8612835715, 190223180840, 4263421511271, 96723482198980, 2216905597676000, 51256802757808320, 1194060413809070710, 27999654303202465310, 660370070571422998410, 15654733143626084944150
COMMENTS
Ninth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 10-Raney sequence. a(n), n>=1, enumerates 10-ary trees (rooted, ordered, incomplete) with n vertices (including the root).
See Graham et al., Hilton and Pedersen, Hoggat and Bicknell, Frey and Sellers references given in A062993. (End) This is instance k = 10 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564 - Wolfdieter Lang, Feb 05 2024
REFERENCES
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
FORMULA
G.f. A(x) satisfies: A = x + A^10.
a(n) = binomial(k*n, n)/((k-1)*n+1), for k=10.
Recurrence: a(0) = 1; a(n) = Sum_{i1+i2+..i10=n-1} a(i1)*a(i2)*...*a(i10) for n>=1. - Robert FERREOL, Apr 01 2015 G.f.: RootOf((_Z^10)*x-_Z+1) (Maple notation, from ECS, see links for A007556). G.f.: hypergeometric([1, 2, 3, 4, 5, 6, 7, 8, 9]/10, [2, 3, 4, 5, 6, 7, 8, 10]/9, (10^10/9^9)*x),
E.g.f.: hypergeometric([1, 2, 3, 4, 5, 6, 7, 8, 9]/10, [2, 3, 4, 5, 6, 7, 8, 9, 10]/9, (10^10/9^9)*x).
For other family members see the crossreferences.
(End)
D-finite with recurrence 81*n*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(9*n+1)*(3*n-2)*(9*n-4)*(9*n-2)*a(n) -800*(10*n-9)*(5*n-4)*(10*n-7)*(5*n-3)*(2*n-1)*(5*n-2)*(10*n-3)*(5*n-1)*(10*n-1)*a(n-1)=0. - R. J. Mathar, Mar 21 2022
EXAMPLE
There are a(2)=10 10-ary trees (vertex degree <=10 and 10 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 10 trees yields 10*10+binomial(10,2)=145=a(3) such trees. - Wolfdieter Lang, Sep 14 2007.
MAPLE
seq(binomial(10*k+1, k)/(9*k+1), k=0..30);
n:=30:G:=series(RootOf(g = 1+x*g^10, g), x=0, n+1):seq(coeff(G, x, k), k=0..n); # Robert FERREOL, Apr 01 2015
MATHEMATICA
a[n_] := Binomial[10n, n]/(9n+1);
CROSSREFS
Related algebraic sequences concerning trees: strictly k-ary trees ( A000108: s=x+s^2, A001263: s=(x, y)+(x, s)+(s, y)+(s, s))), ( A001764: s=x+s^3), ( A002293: s=x+s^4), ( A002294: s=x+s^5), ( A002295: s=x+s^6), ( A002296: s=x+s^7), ( A007556: s=x+s^8), at most k-ary trees ( A001006: s=x+xs+xs^2), ( A036765- A036769, s=x+xs^2....+xs^k, k=3, 4, 5, 6, 7).
AUTHOR
Claude Lenormand (claude.lenormand(AT)free.fr), Mar 05 2001
Array read by antidiagonals giving number of paths up and left from (0,0) to (n,kn) where x/y <= k for all intermediate points.
+10
19
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 12, 14, 1, 1, 1, 5, 22, 55, 42, 1, 1, 1, 6, 35, 140, 273, 132, 1, 1, 1, 7, 51, 285, 969, 1428, 429, 1, 1, 1, 8, 70, 506, 2530, 7084, 7752, 1430, 1, 1, 1, 9, 92, 819, 5481, 23751, 53820, 43263, 4862, 1, 1, 1, 10, 117, 1240
COMMENTS
Also related to dissections of polygons and enumeration of trees.
Number of dissections of a polygon into n (k+2)-gons by nonintersecting diagonals. All dissections are counted separately. See A295260 for nonequivalent solutions up to rotation and reflection. - Andrew Howroyd, Nov 20 2017 Number of rooted polyominoes composed of n (k+2)-gonal cells of the hyperbolic (Euclidean for k=0) regular tiling with Schläfli symbol {k+2,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. For k>0, a stereographic projection of the {k+2,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024
FORMULA
If P(k,x) = Sum_{n>=0} T(n,k)*x^n is the g.f. of column k (k>=0), then P(k,x) = exp(1/(k+1)*(Sum_{j>0} (1/j)*binomial((k+1)*j,j)*x^j)). - Werner Schulte, Oct 13 2015
EXAMPLE
Rows start:
===========================================================
n\k| 0 1 2 3 4 5 6
---|-------------------------------------------------------
0 | 1, 1, 1, 1, 1, 1, 1 ...
1 | 1, 1, 1, 1, 1, 1, 1 ...
2 | 1, 2, 3, 4, 5, 6, 7 ...
3 | 1, 5, 12, 22, 35, 51, 70 ...
4 | 1, 14, 55, 140, 285, 506, 819 ...
5 | 1, 42, 273, 969, 2530, 5481, 10472 ...
6 | 1, 132, 1428, 7084, 23751, 62832, 141778 ...
7 | 1, 429, 7752, 53820, 231880, 749398, 1997688 ...
8 | 1, 1430, 43263, 420732, 2330445, 9203634, 28989675 ...
...
MAPLE
A:= (n, k)-> binomial((k+1)*n, n)/(k*n+1):
MATHEMATICA
T[n_, k_] = Binomial[n(k+1), n]/(k*n+1); Flatten[Table[T[n-k, k], {n, 0, 9}, {k, n, 0, -1}]] (* Jean-François Alcover, Apr 08 2016 *)
PROG
(PARI) T(n, k) = binomial(n*(k+1), n)/(n*k+1); \\ Andrew Howroyd, Nov 20 2017
CROSSREFS
Reflected version of A062993 (which is the main entry).
G.f. satisfies: A(x) = 1 + x*A(x)^5*A(-x)^4.
+10
19
1, 1, 1, 5, 9, 55, 117, 775, 1785, 12350, 29799, 211876, 527085, 3818430, 9706503, 71282640, 184138713, 1366368375, 3573805950, 26735839650, 70625252863, 531838637759, 1416298046436, 10723307329700, 28748759731965, 218658647805780, 589546754316126
COMMENTS
Number of achiral noncrossing partitions composed of n blocks of size 9. - Andrew Howroyd, Feb 08 2024
FORMULA
G.f. satisfies: A(x) = [A(x)*A(-x)] + x*[A(x)*A(-x)]^5.
G.f. satisfies: A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2) where G(x) = 1 + x*G(x)^9 is the g.f. of A062994. a(2n) = binomial(9*n,n)/(8*n+1); a(2n+1) = binomial(9*n+4,n)*5/(8*n+5).
EXAMPLE
G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 9*x^4 + 55*x^5 + 117*x^6 + 775*x^7 +...
Let G(x) = 1 + x*G(x)^9 be the g.f. of A062994, then G(x^2) = A(x)*A(-x) and A(x) = G(x^2) + x*G(x^2)^5 where
G(x) = 1 + x + 9*x^2 + 117*x^3 + 1785*x^4 + 29799*x^5 + 527085*x^6 +...
G(x)^5 = 1 + 5*x + 55*x^2 + 775*x^3 + 12350*x^4 + 211876*x^5 +...
MATHEMATICA
terms = 25;
A[_] = 1; Do[A[x_] = 1 + x A[x]^5 A[-x]^4 + O[x]^terms // Normal, {terms}];
PROG
(PARI) {a(n)=my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^5*subst(A^4, x, -x)); polcoef(A, n)}
(PARI) {a(n)=my(m=n\2, p=4*(n%2)+1); binomial(9*m+p-1, m)*p/(8*m+p)}
Triangle of coefficients of polynomials P(n,t) related to the Mittag-Leffler function, where P(n,t) = Product_{k=0..n-2} n*t-k.
+10
18
1, 0, 2, 0, -3, 9, 0, 8, -48, 64, 0, -30, 275, -750, 625, 0, 144, -1800, 7560, -12960, 7776, 0, -840, 13426, -77175, 204085, -252105, 117649, 0, 5760, -112896, 831488, -3010560, 5734400, -5505024, 2097152, 0, -45360, 1058508, -9573228
COMMENTS
Second column (unsigned) 2, 3, 8, 30, 144, ... is A001048. Diagonal 1, 2, 9, 64, 625, 7776, ... is A000169.
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998
FORMULA
P(n,t) = (n-1)!*binomial(n*t, n-1).
E.g.f. (with constant term 1): B_t(x) = Sum_{n >= 0} 1/(n*t + 1)*binomial(n*t + 1,n)*x^n = 1 + x + 2*t*x^2/2! + 3*t(3*t - 1)*x^3/3! + 4*t*(4*t - 1)*(4*t - 2)*x^4/4! + ... is the generalized binomial series of Lambert. See Graham et al., Section 5.4 and Section 7.5.
In the notation of the Bala link, B_t(x) = I^t(1 + x) where I^t is a fractional inversion operator. B_(1+t)(x) is the e.g.f. for A260687. B_t(x) = 1 + x*B_t(x)^t.
For complex r, B_t(x)^r = Sum_{n >= 0} r/(n*t + r)*binomial(n*t + r,n)*x^n.
log (B_t(x)) = Sum_{n >= 1} 1/(n*t)*binomial(n*t,n)*x^n.
B_2(x) is the o.g.f. for the Catalan numbers A000108. B_t(x) for t = 3,4,5,... gives the o.g.f. for various Fuss-Catalan sequences. See the cross references. (End)
EXAMPLE
Triangle begins :
1;
0, 2;
0, -3, 9;
0, 8, -48, 64;
0, -30, 275, -750, 625;
0, 144, -1800, 7560, -12960, 7776;
...
MATHEMATICA
P[n_, t_] := Product[n*t - k, {k, 0, n-2}]; row[n_] := CoefficientList[P[n, t], t]; Table[row[n], {n, 1, 10}] // Flatten
CROSSREFS
Cf. A000169, A001048, A156136, A000108 (B_2(x)), A001764 (B_3(x)), A002293 (B_4(x)), A002294 (B_5(x)), A002295 (B_6(x)), A002296 (B_7(x)), A007556 (B_8(x)), A062994 (B_9(x)), A059968 (B_10(x)), A230388 (B_11(x)), A139526, A260687.
A triangle (lower triangular matrix) composed of Pfaff-Fuss (or Raney) sequences.
+10
17
1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 14, 12, 4, 1, 1, 42, 55, 22, 5, 1, 1, 132, 273, 140, 35, 6, 1, 1, 429, 1428, 969, 285, 51, 7, 1, 1, 1430, 7752, 7084, 2530, 506, 70, 8, 1, 1, 4862, 43263, 53820, 23751, 5481, 819, 92, 9
COMMENTS
The column sequences (without leading zeros) appear in eq.(7.66), p. 347 of the Graham et al. reference, in Th. 0.3, p. 66, of Hilton and Pedersen reference, as first columns of the S-triangles in the Hoggatt and Bicknell reference and in eq. 5 of the Frey and Sellers reference. They are also called m-Raney (here m=k+2) or Fuss-Catalan sequences (see Graham et al. for reference). For the history and the name Pfaff-Fuss see Brown reference, p. 975. PF(n,m) := binomial(m*n+1,n)/(m*n+1), m >= 2.
Also called generalized Catalan numbers.
REFERENCES
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994.
FORMULA
a(n, k) = binomial((k+2)*(n-k), n-k)/((k+1)*(n-k)+1) = PF(n-k, k+2) if n-k >= 0, otherwise 0.
G.f. for column k: A(k, x) := x^k*RootOf(_Z^(k+2)*x-_Z+1) (Maple notation, from ECS, see links for column sequences and Graham et al. reference eq.(5.59) p. 200 and p. 349).
EXAMPLE
The triangle a(n, k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 1
1: 1 1
2: 2 1 1
3: 5 3 1 1
4: 14 12 4 1 1
5: 42 55 22 5 1 1
6: 132 273 140 35 6 1 1
7: 429 1428 969 285 51 7 1 1
8: 1430 7752 7084 2530 506 70 8 1 1
9: 4862 43263 53820 23751 5481 819 92 9 1 1
10: 16796 246675 420732 231880 62832 10472 1240 117 10 1 1
MATHEMATICA
a[n_, k_] = Binomial[(k+2)*(n-k), n-k]/((k+1)*(n-k) + 1);
Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 53]]
a(n) = 10*binomial(9*n + 10, n)/(9*n + 10).
+10
14
1, 10, 135, 2100, 35475, 632502, 11714745, 223198440, 4346520750, 86128357150, 1731030945644, 35202562937100, 723029038312230, 14976976398326250, 312522428615310000, 6563314391270476752, 138617681440915119975, 2942332729799060033100, 62735156704285184848950
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 9, r = 10.
FORMULA
G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 9, r = 10.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^10), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11. A(x)^(1/10) is the o.g.f. for A062994. (End) D-finite with recurrence: 128*n*(8*n+3)*(4*n+3)*(8*n+9)*(2*n+1)*(8*n+7)*(4*n+5)*(8*n+5)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
MATHEMATICA
Table[10 Binomial[9 n + 10, n]/(9 n + 10), {n, 0, 30}]
PROG
(PARI) a(n) = 10*binomial(9*n+10, n)/(9*n+10);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/10))^10+x*O(x^n)); polcoeff(B, n)}
(Magma) [10*Binomial(9*n+10, n)/(9*n+10): n in [0..30]];
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