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Search: a006631 -id:a006631
Displaying 1-6 of 6 results found. page 1
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A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k. +10
9
1, 2, 1, 7, 4, 1, 30, 18, 6, 1, 143, 88, 33, 8, 1, 728, 455, 182, 52, 10, 1, 3876, 2448, 1020, 320, 75, 12, 1, 21318, 13566, 5814, 1938, 510, 102, 14, 1, 120175, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 690690, 444015, 197340, 70840, 21252, 5313, 1078, 168, 18, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
With offset 0, Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A006013. - Philippe Deléham, Jan 23 2010
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, The second production matrix of a Riordan array, arXiv:2011.13985 [math.CO], 2020.
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.
FORMULA
T(n, k) = 2*k*binomial(3n-k, n-k)/(3n-k).
G.f.: 1/(1-t*z*g^2), where g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z) is the g.f. of the sequence A001764.
T(n, k) = Sum_{j>=1} j*T(n-1, k-2+j). - Philippe Deléham, Sep 14 2005
With offset 0, T(n,k) = ((n+1)/(k+1))*binomial(3n-k+1, n-k). - Philippe Deléham, Jan 23 2010
From Gary W. Adamson, Jul 07 2011: (Start)
Let M = the production matrix
2, 1;
3, 2, 1;
4, 3, 2, 1;
5, 4, 3, 2, 1;
...
Top row of M^(n-1) generates n-th row terms of triangle A092276. Leftmost terms of each row = A006013 starting (1, 2, 7, 30, 143, ...). (End)
Working with an offset of 0, the inverse array is the Riordan array ((1 - x)^2, x*(1 - x)^2). - Peter Bala, Apr 30 2024
EXAMPLE
Triangle begins:
1;
2, 1;
7, 4, 1;
30, 18, 6, 1;
143, 88, 33, 8, 1;
728, 455, 182, 52, 10, 1;
3876, 2448, 1020, 320, 75, 12, 1;
...
Top row of M^3 = (30, 18, 6, 1)
MAPLE
T := proc(n, k) if k=n then 1 else 2*k*binomial(3*n-k, n-k)/(3*n-k) fi end: seq(seq(T(n, k), k=1..n), n=1..11);
MATHEMATICA
t[n_, n_] = 1; t[n_, k_] := 2*k*Binomial[3*n-k, n-k]/(3*n-k); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2012, after Maple *)
PROG
(PARI) T(n, k) = 2*k*binomial(3*n-k, n-k)/(3*n-k); \\ Andrew Howroyd, Nov 06 2017
CROSSREFS
Row sums give sequence A001764.
Columns 1..5 are A006013, A006629, A006630, A006631, A233657.
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Feb 24 2004
STATUS
approved
A006633 Expansion of hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x).
(Formerly M4230) 		+10
7
1, 6, 39, 272, 1995, 15180, 118755, 949344, 7721604, 63698830, 531697881, 4482448656, 38111876530, 326439471960, 2814095259675, 24397023508416, 212579132600076, 1860620845932216, 16351267454243260, 144222309948974400, 1276307560533365955, 11329053395044653180 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
From generalized Catalan numbers.
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..1000
H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
Simon Plouffe, Approximations of generating functions and a few conjectures, Master's Thesis, arXiv:0911.4975 [math.NT], 2009.
FORMULA
O.g.f.: hypergeom_4F3([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x). - Simon Plouffe, Master's Thesis, UQAM 1992
a(n) = 2*binomial(4*n + 5, n) / (n+2). - Bruno Berselli, Jan 18 2014
a(n) = (n+1) * A000260(n+1). - F. Chapoton, Feb 22 2024
MAPLE
gf := hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x):
ser := series(gf, x, 22): seq(coeff(ser, x, n), n = 0..21); # Peter Luschny, Feb 22 2024
MATHEMATICA
A006633[n_] := 2*Binomial[4*n+5, n]/(n+2);
Array[A006633, 25, 0] (* Paolo Xausa, Feb 25 2024 *)
CROSSREFS
Cf. A006631, A006632, A006634, A006635, A000027, A000260.
KEYWORD
nonn
AUTHOR
Simon Plouffe
EXTENSIONS
New name by using a formula from the author by Peter Luschny, Feb 24 2024
STATUS
approved
A110616 A convolution triangle of numbers based on A001764. +10
5
1, 1, 1, 3, 2, 1, 12, 7, 3, 1, 55, 30, 12, 4, 1, 273, 143, 55, 18, 5, 1, 1428, 728, 273, 88, 25, 6, 1, 7752, 3876, 1428, 455, 130, 33, 7, 1, 43263, 21318, 7752, 2448, 700, 182, 42, 8, 1, 246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Reflected version of A069269. - Vladeta Jovovic, Sep 27 2006
With offset 1 for n and k, T(n,k) = number of Dyck paths of semilength n for which all descents are of even length (counted by A001764) with no valley vertices at height 1 and with k returns to ground level. For example, T(3,2)=2 counts U^4 D^4 U^2 D^2, U^2 D^2 U^4 D^4 where U=upstep, D=downstep and exponents denote repetition. - David Callan, Aug 27 2009
Riordan array (f(x), x*f(x)) with f(x) = (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))). - Philippe Deléham, Jan 27 2014
Antidiagonals of convolution matrix of Table 1.4, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019
LINKS
Table of n, a(n) for n=0..54.
Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
Sheng-Liang Yang, LJ Wang, Taylor expansions for the m-Catalan numbers, Australasian Journal of Combinatorics, Volume 64(3) (2016), Pages 420-431.
FORMULA
T(n, k) = Sum_{j>=0} T(n-1, k-1+j)*A000108(j); T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n.
G.f.: 1/(1 - x*y*TernaryGF) = 1 + (y)x + (y+y^2)x^2 + (3y+2y^2+y^3)x^3 +... where TernaryGF = 1 + x + 3x^2 + 12x^3 + ... is the GF for A001764. - David Callan, Aug 27 2009
T(n, k) = ((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1). - Vladimir Kruchinin, Nov 01 2011
EXAMPLE
Triangle begins:
1;
1, 1;
3, 2, 1;
12, 7, 3, 1;
55, 30, 12, 4, 1;
273, 143, 55, 18, 5, 1;
1428, 728, 273, 88, 25, 6, 1;
7752, 3876, 1428, 455, 130, 33, 7, 1;
43263, 21318, 7752, 2448, 700, 182, 42, 8, 1;
246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1;
...
MATHEMATICA
Table[(k + 1) Binomial[3 n - 2 k, 2 n - k]/(2 n - k + 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 28 2017 *)
PROG
(Maxima) T(n, k):=((k+1)*binomial(3*n-2*k, 2*n-k))/(2*n-k+1); // Vladimir Kruchinin, Nov 01 2011
CROSSREFS
Successive columns: A001764, A006013, A001764, A006629, A102893, A006630, A102594, A006631; row sums: A098746; see also A092276.
KEYWORD
nonn,tabl
AUTHOR
Philippe Deléham, Sep 14 2005, Jun 15 2007
STATUS
approved
A069269 Second level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 3rd level is A069270). +10
3
1, 1, 1, 1, 2, 3, 1, 3, 7, 12, 1, 4, 12, 30, 55, 1, 5, 18, 55, 143, 273, 1, 6, 25, 88, 273, 728, 1428, 1, 7, 33, 130, 455, 1428, 3876, 7752, 1, 8, 42, 182, 700, 2448, 7752, 21318, 43263, 1, 9, 52, 245, 1020, 3876, 13566, 43263, 120175, 246675, 1, 10, 63, 320, 1428 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
For the m-th level generalization of Catalan triangle T(n,k) = C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n >= k+m: T(n,k) = T(n-m+1,k+1) - T(n-m,k+1); and T(n,n) = T(n+m-1,n-1) = C((m+1)n,n)/(mn+1).
Reflected version of A110616. - Philippe Deléham, Jun 15 2007
With offset 1 for n and k, T(n,k) is (conjecturally) the number of permutations of [n] that avoid the patterns 4-2-3-1 and 4-2-5-1-3 and for which the last ascent ends at position k (k=1 if there are no ascents). For example, T(4,1) = 1 counts 4321; T(4,2) = 3 counts 1432, 2431, 3421; T(4,3) = 7 counts 1243, 1342, 2143, 2341, 3142, 3241, 4132. - David Callan, Jul 22 2008
Row sums appear to be in A098746. - R. J. Mathar, May 30 2014
LINKS
Table of n, a(n) for n=0..59.
M. H. Albert et al., Restricted permutations and queue jumping, Discrete Math., 287 (2004), 129-133.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 18.
FORMULA
T(n, k) = C(n+2k, k)*(n-k+1)/(n+k+1).
For n >= k+2: T(n, k) = T(n-1, k+1) - T(n-2, k+1).
T(n, n) = T(n+1, n-1) = C(3n, n)/(2n+1).
EXAMPLE
Rows start
1;
1, 1;
1, 2, 3;
1, 3, 7, 12;
1, 4, 12, 30, 55;
CROSSREFS
Columns include A000012, A000027, A055998.
Right-hand diagonals include A001764, A006013, A006629, A006630, A006631.
Cf. triangles A007318, A009766, A069270.
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Mar 12 2002
STATUS
approved
A230547 a(n) = 3*binomial(3*n+9, n)/(n+3). +10
3
1, 9, 63, 408, 2565, 15939, 98670, 610740, 3786588, 23535820, 146710476, 917263152, 5752004349, 36174046743, 228124619100, 1442387942520, 9142452842985, 58083251802345, 369816259792035, 2359448984037600 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=9.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
FORMULA
G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=3, r=9.
MATHEMATICA
Table[9 Binomial[3 n + 9, n]/(3 n + 9), {n, 0, 30}]
PROG
(PARI) a(n) = 9*binomial(3*n+9, n)/(3*n+9);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/9))^9+x*O(x^n)); polcoeff(B, n)}
(Magma) [9*Binomial(3*n+9, n)/(3*n+9): n in [0..30]];
CROSSREFS
Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A233657.
KEYWORD
nonn
AUTHOR
Tim Fulford, Oct 23 2013
STATUS
approved
A233657 a(n) = 10 * binomial(3*n+10,n)/(3*n+10). +10
3
1, 10, 75, 510, 3325, 21252, 134550, 848250, 5340060, 33622600, 211915132, 1337675430, 8458829925, 53591180360, 340185835500, 2163581913780, 13786238414025, 88004926973250, 562763873596575, 3604713725613000, 23126371951808268, 148594788106641360 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=10.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, arXiv preprint arXiv:1711.10325 [math.CO], 2017-2019.
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955 (2010).
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
FORMULA
G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=3, r=10.
+2*n*(n+5)*(2*n+9)*a(n) -3*(3*n+7)*(n+3)*(3*n+8)*a(n-1)=0. - R. J. Mathar, Feb 16 2018
E.g.f.: F([10/3, 11/3, 4], [1, 11/2, 6], 27*x/4), where F is the generalized hypergeometric function. - Stefano Spezia, Oct 08 2019
MAPLE
A233657:=n->10*binomial(3*n+10, n)/(3*n+10): seq(A233657(n), n=0..20); # Wesley Ivan Hurt, Oct 10 2014
MATHEMATICA
Table[10 Binomial[3 n + 10, n]/(3 n + 10), {n, 0, 30}]
PROG
(PARI) a(n) = 10*binomial(3*n+10, n)/(3*n+10);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/10))^10+x*O(x^n)); polcoeff(B, n)}
(Magma) [10*Binomial(3*n+10, n)/(3*n+10): n in [0..30]];
CROSSREFS
Column 5 of A092276.
Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A230547.
KEYWORD
nonn,easy
AUTHOR
Tim Fulford, Dec 14 2013
STATUS
approved
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Last modified August 29 21:13 EDT 2024. Contains 375518 sequences. (Running on oeis4.)