Displaying 1-10 of 13 results found.
Triple factorial numbers (3*n-2)!!! with leading 1 added.
(Formerly M3627)
+10
119
1, 1, 4, 28, 280, 3640, 58240, 1106560, 24344320, 608608000, 17041024000, 528271744000, 17961239296000, 664565853952000, 26582634158080000, 1143053268797440000, 52580450364682240000, 2576442067869429760000, 133974987529210347520000, 7368624314106569113600000
COMMENTS
a(n) is the number of increasing quaternary trees on n vertices. (See A001147 for ternary and A000142 for binary trees.) - David Callan, Mar 30 2007 a(n) is the product of the positive integers k <= 3*n that have k modulo 3 = 1. - Peter Luschny, Jun 23 2011 a(n) is the number of generalized permutations of length n related to the degenerate Eulerian numbers (see arXiv:2007.13205), cf. A336633. - Orli Herscovici, Jul 28 2020
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = Product_{k=0..n-1} (3*k + 1).
a(n) = (3*n - 2)!!!, n >= 1, and a(0) = 1.
E.g.f.: (1-3*x)^(-1/3).
a(n) ~ sqrt(2*Pi)/Gamma(1/3)*n^(-1/6)*(3*n/e)^n*(1 - (1/36)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
a(n) = 3^n*Pochhammer(1/3, n).
a(n) = n! *( Sum_{m=1..n} (m/n)*Sum_{k=1..n-m} (binomial(k, n-m-k) * (-1/3)^(n-m-k)*binomial(k+n-1,n-1) + 1 ), n>1. - Vladimir Kruchinin, Aug 09 2010 a(n) = upper left term in M^n, M = a variant of Pascal (1,3) triangle (Cf. A095660); as an infinite square production matrix: 1, 3, 0, 0, 0,...
1, 4, 3, 0, 0,...
1, 5, 7, 3, 0,...
...
a(n+1) = sum of top row terms of M^n. (End)
a(n) = (-2)^n*Sum_{k=0..n} (3/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012 G.f.: 1/Q(0) where Q(k) = 1 - x*(3*k+1)/( 1 - x*(3*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 21 2013 G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013 E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - x*(3*k+1)/(x*(3*k+1) + (k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013 Let D(x) = 1/sqrt(1 - 2*x) be the e.g.f. for the sequence of double factorial numbers A001147. Then the e.g.f. A(x) for the triple factorial numbers satisfies D( Integral_{t=0..x} A(t) dt ) = A(x). Cf. A007696 and A008548. - Peter Bala, Jan 02 2015 O.g.f.: hypergeom([1, 1/3], [], 3*x). - Peter Luschny, Oct 08 2015 a(n) = sigma[3,1]^{(n)}_n, n >= 0, with the elementary symmetric function of degree n in the n numbers 1, 4, 7, ..., 1+3*(n-1), with sigma[3,1]^{(n)}_0 := 1. See the first formula. - Wolfdieter Lang, May 29 2017 a(n) = (-1)^n / A008544(n), 0 = a(n)*(+3*a(n+1) -a(n+2)) +a(n+1)*a(n+1) for all n in Z. - Michael Somos, Sep 30 2018 D-finite with recurrence: a(n) +(-3*n+2)*a(n-1)=0, n>=1. - R. J. Mathar, Feb 14 2020 Sum_{n>=1} 1/a(n) = (e/9)^(1/3) * (Gamma(1/3) - Gamma(1/3, 1/3)). - Amiram Eldar, Jun 29 2020
EXAMPLE
G.f. = 1 + x + 4*x^2 + 28*x^3 + 280*x^4 + 3640*x^5 + 58240*x^6 + ...
a(3) = 28 and a(4) = 280; with top row of M^3 = (28, 117, 108, 27), sum = 280.
MATHEMATICA
a[ n_] := If[ n < 0, 1 / Product[ k, {k, - 2, 3 n - 1, -3}],
Product[ k, {k, 1, 3 n - 2, 3}]]; (* Michael Somos, Oct 14 2011 *) FoldList[Times, 1, Range[1, 100, 3]] (* Harvey P. Dale, Jul 05 2013 *) Range[0, 19]! CoefficientList[Series[((1 - 3 x)^(-1/3)), {x, 0, 19}], x] (* Vincenzo Librandi, Oct 08 2015 *)
PROG
(Maxima) a(n):=if n=1 then 1 else (n)!*(sum(m/n*sum(binomial(k, n-m-k)*(-1/3)^(n-m-k)* binomial (k+n-1, n-1), k, 1, n-m), m, 1, n)+1); \\ Vladimir Kruchinin, Aug 09 2010 (PARI) {a(n) = if( n<0, (-1)^n / prod(k=0, -1-n, 3*k + 2), prod(k=0, n-1, 3*k + 1))}; /* Michael Somos, Oct 14 2011 */ (PARI) x='x+O('x^33); Vec(serlaplace((1-3*x)^(-1/3))) /* Joerg Arndt, Apr 24 2011 */ (Sage)
def A007559(n) : return mul(j for j in range(1, 3*n, 3)) (Haskell)
a007559 n = a007559_list !! n
a007559_list = scanl (*) 1 a016777_list
(Magma)
b:= func< n | (n lt 2) select n else (3*n-2)*Self(n-1) >;
(GAP) List([0..20], n-> Product([0..n-1], k-> 3*k+1 )); # G. C. Greubel, Aug 20 2019
CROSSREFS
a(n)= A035469(n, 1), n >= 1, (first column of triangle A035469(n, m)).
Triple factorial numbers: Product_{k=0..n-1} (3*k+2).
+10
80
1, 2, 10, 80, 880, 12320, 209440, 4188800, 96342400, 2504902400, 72642169600, 2324549427200, 81359229952000, 3091650738176000, 126757680265216000, 5577337931669504000, 262134882788466688000
COMMENTS
a(n-1), n >= 1, enumerates increasing plane (aka ordered) trees with n vertices (one of them a root labeled 1) where each vertex with outdegree r >= 0 comes in r+1 types (like an (r+1)-ary vertex). See the increasing tree comments under A004747. - Wolfdieter Lang, Oct 12 2007 An example for the case of 3 vertices is shown below. For the enumeration of non-plane trees of this type see A029768. - Peter Bala, Aug 30 2011 a(n) is the product of the positive integers k <= 3*n that have k modulo 3 = 2. - Peter Luschny, Jun 23 2011 The Mathar conjecture is true. Generally from the factorial form, the last term is the "extra" product beyond the prior term, from k=n-1 and 3k+2 evaluates to 3*(n-1)+2 = 3n-1, yielding a(n) = a(n-1)*(3n-1) (eqn1). Similarly, a(n) = a(n-2)*(3n-1)*(3(n-2)+2) = a(n-2)*(3n-1)*(3n-4) (eqn2) and a(n) = a(n-3)*(3n-1)*(3n-4)*(3*(n-2)+2) = a(n-3)*(3n-1)*(3n-4)*(3n-7) (eqn3). We equate (eqn2) and (eqn3) to get a(n-2)*(3n-1)*(3n-4) = a(n-3*(3n-1)*(3n-4)*(3n-7) or a(n-2)+(7-3n)*a(n-3) = 0 (eqn4). From (eqn1) we have a(n)+(1-3n)*a(n-1) = 0 (eqn5). Combining (eqn4) and (eqn5) yields a(n)+(1-3n)*a(n-1)+a(n-2)+(7-3n)*a(n-3) = 0. - Bill McEachen, Jan 01 2016 a(n-1), n>=1, is the dimension of the n-th component of the operad encoding the multilinearization of the following identity in nonassociative algebras: s*(a,a,b)-(s+t)*(a,b,a)+t*(b,a,a)=0, for any given pair of scalars (s,t). Here (a,b,c) is the associator (ab)c-a(bc). This is proved in the referenced article on associator dependent algebras by Bremner and me. - Vladimir Dotsenko, Mar 22 2022
FORMULA
a(n) = Product_{k=0..n-1} (3*k+2) = A007661(3*n-1) (with A007661(-1) = 1). E.g.f.: (1-3*x)^(-2/3).
a(n) = 2* A034000(n), n >= 1, a(0) = 1. a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(2/3)^-1*n^(1/6)*3^n*e^-n*n^n*{1 - 1/36*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 22 2001
From Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003: (Start)
a(n) = 3^n*Pochhammer(2/3, n) = 3^n*Gamma(n+2/3)/Gamma(2/3). (End)
Let T = A094638 and c(t) = column vector(1, t, t^2, t^3, t^4, t^5,...), then A008544 = unsigned [ T * c(-3) ] and the list partition transform A133314 of [1,T * c(-3)] gives [1,T * c(3)] with all odd terms negated, which equals a signed version of A007559; i.e., LPT[(1,signed A008544)] = signed A007559. Also LPT[ A007559] = (1,- A008544) and e.g.f. [1,T * c(t)] = (1-x*t)^(-1/t) for t = 3 or -3. Analogous results hold for the double factorial, quadruple factorial and so on. - Tom Copeland, Dec 22 2007 G.f.: 1/(1-2x/(1-3x/(1-5x/(1-6x/(1-8x/(1-9x/(1-11x/(1-12x/(1-...))))))))) (continued fraction). - Philippe Deléham, Jan 08 2012 a(n) = (-1)^n*Sum_{k=0..n} 3^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012 G.f.: 1/Q(0) where Q(k) = 1 - x*(3*k+2)/(1 - x*(3*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013 G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(3*k+2)/(x*(3*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 25 2013 D-finite with recurrence: a(n) = (9*(n-2)*(n-1)+2)*a(n-2) + 4*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 09 2013 a(n) = n!*Sum_{k=floor(n/2)..n} binomial(k,n-k)*binomial(n+k,k)*3^(-n+k)*(-1)^(n-k). - Vladimir Kruchinin, Sep 28 2013 Recurrence equation: a(n) = 3*a(n-1) + (3*n - 4)^2*a(n-2) with a(0) = 1 and a(1) = 2. A024396 satisfies the same recurrence (but with different initial conditions). This observation leads to a continued fraction expansion for the constant A193534 due to Euler. - Peter Bala, Feb 20 2015 G.f.: Hypergeometric2F0(1, 2/3; -; 3*x). - G. C. Greubel, Mar 31 2019 D-finite with recurrence: a(n) + (-3*n+1)*a(n-1)=0. - R. J. Mathar, Jan 17 2020 G.f.: 1/(1-2*x-6*x^2/(1-8*x-30*x^2/(1-14*x-72*x^2/(1-20*x-132*x^2/(1-...))))) (Jacobi continued fraction). - Nikolaos Pantelidis, Feb 28 2020 G.f.: 1/G(0), where G(k) = 1 - (6*k+2)*x - 3*(k+1)*(3*k+2)*x^2/G(k+1). - Nikolaos Pantelidis, Feb 28 2020 Sum_{n>=0} 1/a(n) = 1 + (e/3)^(1/3) * (Gamma(2/3) - Gamma(2/3, 1/3)). - Amiram Eldar, Mar 01 2022
EXAMPLE
a(2) = 10 from the described trees with 3 vertices: there are three trees with a root vertex (label 1) with outdegree r=2 (like the three 3-stars each with one different ray missing) and the four trees with a root (r=1 and label 1) a vertex with (r=1) and a leaf (r=0). Assigning labels 2 and 3 yields 2*3+4=10 such trees.
a(2) = 10. The 10 possible plane increasing trees on 3 vertices, where vertices of outdegree 1 come in 2 colors (denoted a or b) and vertices of outdegree 2 come in 3 colors (a, b or c), are:
.
1a 1b 1a 1b 1a 1b 1c
| | | | / \ / \ / \
2a 2b 2b 2a 2 3 2 3 2 3
| | | |
3 3 3 3 1a 1b 1c
/ \ / \ / \
3 2 3 2 3 2
MAPLE
a := n -> mul(3*k-1, k = 1..n);
MATHEMATICA
k = 3; b[1]=2; b[n_]:= b[n] = b[n-1]+k; a[0]=1; a[1]=2; a[n_]:= a[n] = a[n-1]*b[n]; Table[a[n], {n, 0, 20}] (* Roger L. Bagula, Sep 17 2008 *) Table[3^n*Pochhammer[2/3, n], {n, 0, 20}] (* G. C. Greubel, Mar 31 2019 *)
PROG
(PARI) a(n) = prod(k=0, n-1, 3*k+2 );
(PARI) vector(20, n, n--; round(3^n*gamma(n+2/3)/gamma(2/3))) \\ G. C. Greubel, Mar 31 2019 (Sage)
@CachedFunction
(Sage) [3^n*rising_factorial(2/3, n) for n in (0..20)] # G. C. Greubel, Mar 31 2019 (Haskell)
a008544 n = a008544_list !! n
a008544_list = scanl (*) 1 a016789_list
(Maxima)
a(n):=((n)!*sum(binomial(k, n-k)*binomial(n+k, k)*3^(-n+k)*(-1)^(n-k), k, floor(n/2), n)); /* Vladimir Kruchinin, Sep 28 2013 */ (Magma) [Round((Gamma(2*n-5/3)/Gamma(n-5/6)*Gamma(2/3)/Gamma(5/6) )/ Sqrt(3)*3^n/4^(n-1)): n in [1..20]]; // Vincenzo Librandi, Feb 21 2015 (Magma) [Round(3^n*Gamma(n+2/3)/Gamma(2/3)): n in [0..20]]; // G. C. Greubel, Mar 31 2019
AUTHOR
Joe Keane (jgk(AT)jgk.org)
Triple factorial numbers: (3n)!!! = 3^n*n!.
+10
50
1, 3, 18, 162, 1944, 29160, 524880, 11022480, 264539520, 7142567040, 214277011200, 7071141369600, 254561089305600, 9927882482918400, 416971064282572800, 18763697892715776000, 900657498850357248000, 45933532441368219648000, 2480410751833883860992000
COMMENTS
For n >= 1 a(n) is the order of the wreath product of the symmetric group S_n and the elementary Abelian group (C_3)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001
Laguerre transform of double factorials 2^n*n! = A000165(n). - Paul Barry, Aug 08 2008 For positive n, a(n) equals the permanent of the n X n matrix consisting entirely of 3's. - John M. Campbell, May 26 2011 a(n) is the product of the positive integers <= 3*n that are multiples of 3. - Peter Luschny, Jun 23 2011
FORMULA
a(n) = 3^n*n!.
a(n) = Product_{k=1..n} 3*k.
E.g.f.: 1/(1-3*x).
a(n) = Sum_{k=0..n} C(n,k)*(n!/k!)*2^k*k!. - Paul Barry, Aug 08 2008 G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 6*x*(k+1)/(6*x*(k+1) - 1 + 6*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013 G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k+3)/(x*(3*k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013 G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(2*k+1) - 9*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013 Sum_{n>=0} 1/a(n) = e^(1/3) ( A092041). Sum_{n>=0} (-1)^n/a(n) = e^(-1/3) ( A092615). (End)
MAPLE
with(combstruct):ZL:=[T, {T=Union(Z, Prod(Epsilon, Z, T), Prod(T, Z, Epsilon), Prod(T, Z))}, labeled]:seq(count(ZL, size=i)/i, i=1..17); # Zerinvary Lajos, Dec 16 2007
MATHEMATICA
Join[{1}, FoldList[Times, 3*Range[20]]] (* Harvey P. Dale, Feb 10 2019 *) Table[Times@@Range[3n, 1, -3], {n, 0, 20}] (* Harvey P. Dale, Apr 14 2023 *)
PROG
(PARI) a(n)=3^n*n!;
(PARI) a(n)=prod(k=1, n, 3*k );
(SageMath)
def A032031(n) : return mul(j for j in range(3, 3*(n+1), 3)) (Haskell)
a032031 n = a032031_list !! n
a032031_list = scanl (*) 1 $ tail a008585_list
Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.
+10
38
1, 4, 1, 28, 12, 1, 280, 160, 24, 1, 3640, 2520, 520, 40, 1, 58240, 46480, 11880, 1280, 60, 1, 1106560, 987840, 295960, 40040, 2660, 84, 1, 24344320, 23826880, 8090880, 1296960, 109200, 4928, 112, 1, 608608000, 643843200
COMMENTS
Previous name was: Triangle of numbers related to triangle A035529; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297 and A035342. a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing quartic (4-ary) trees. Proof based on the a(n,m) recurrence. See a D. Callan comment on the m=1 case A007559. See also the F. Bergeron et al. reference, especially Table 1, first row and Example 1 for the e.g.f. for m=1. - Wolfdieter Lang, Sep 14 2007
REFERENCES
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, in Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1922, pp. 24-48.
FORMULA
a(n, m) = Sum_{j=m..n} | A051141(n, j)|*S2(j, m) (matrix product), with S2(j, m):= A008277(j, m) (Stirling2 triangle). Priv. comm. to Wolfdieter Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342. a(n, m) = n!* A035529(n, m)/(m!*3^(n-m)); a(n+1, m) = (3*n+m)*a(n, m) + a(n, m-1), n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0, a(1, 1)=1; E.g.f. of m-th column: ((-1+(1-3*x)^(-1/3))^m)/m!.
E.g.f.: G(x,t) = exp(t*A(x)) = 1 + t*x + (4*t+t^2)*x^2/2! + (28*t + 12*t^2 + t^3)*x^3/3! + ..., where A(x) = -1 + (1-3*x)^(-1/3) satisfies the autonomous differential equation A'(x) = (1+A(x))^4.
The generating function G(x,t) satisfies the partial differential equation t*(dG/dt+G) = (1-3*x)*dG/dx, from which follows the recurrence given above.
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^4*d/dx. Cf. A008277 (D = (1+x)*d/dx), A105278 (D = (1+x)^2*d/dx), A035342 (D = (1+x)^3*d/dx) and A049029 (D = (1+x)^5*d/dx). (End)
Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k>=0} k*(k+3)*(k+6)*...*(k+3*(n-1))*x^k/k!. - Peter Bala, Jun 23 2014
EXAMPLE
Triangle starts:
{1}
{4, 1}
{28, 12, 1}
{280, 160, 24, 1}
{3640, 2520, 520, 40, 1}
MATHEMATICA
a[n_, m_] /; n >= m >= 1 := a[n, m] = (3(n-1) + m)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* Jean-François Alcover, Jul 22 2011 *) rows = 9;
a[n_, m_] := BellY[n, m, Table[Product[3k+1, {k, 0, j}], {j, 0, rows}]];
PROG
(Sage) # uses[bell_matrix from A264428] # Adds a column 1, 0, 0, 0, ... at the left side of the triangle.
CROSSREFS
a(n, m)=: S2(4, n, m) is the fourth triangle of numbers in the sequence S2(1, n, m) := A008277(n, m) (Stirling 2nd kind), S2(2, n, m) := A008297(n, m) (Lah), S2(3, n, m) := A035342(n, m). a(n, 1)= A007559(n).
Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).
+10
21
1, -2, 1, 8, -6, 1, -48, 44, -12, 1, 384, -400, 140, -20, 1, -3840, 4384, -1800, 340, -30, 1, 46080, -56448, 25984, -5880, 700, -42, 1, -645120, 836352, -420224, 108304, -15680, 1288, -56, 1, 10321920, -14026752, 7559936, -2153088, 359184, -36288, 2184, -72, 1
COMMENTS
T(n,m) = R_n^m(a=0,b=2) in the notation of the given reference.
Exponential Riordan array [1/(1+2x),log(1+2x)/2]. The unsigned triangle is [1/(1-2x),log(1/sqrt(1-2x))]. - Paul Barry_, Apr 29 2009
The n-th row is related to the expansion of z^(-2n)*(z^3 d/dz)^n in polynomials of the Euler operator D=(z d/dz). E.g., z^(-6)(z^3 d/dz)^3 = D^3 + 6 D^2 + 8 D. See Copeland link for relations to Bell / Exponential / Touchard polynomial operators. - Tom Copeland, Nov 14 2013 Also the Bell transform of the double factorial of even numbers A000165 except that the values are unsigned and in addition a first column (1,0,0 ...) is added on the left side of the triangle. For the Bell transform of the double factorial of odd numbers A001147 see A132062. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015 The signed triangle is also the inverse Bell transform of A000079 (see Luschny link). - John Keith, Nov 24 2020
FORMULA
T(n, m) = T(n-1, m-1) - 2*(n-1)*T(n-1, m), n >= m >= 1; T(n, m) := 0, n<m; T(n, 0) := 0, T(1, 1)=1.
E.g.f. for m-th column of signed triangle: (((log(1+2*x))/2)^m)/m!.
E.g.f.: (1+2*x)^(y/2). O.g.f. for n-th row of signed triangle: Sum_{m=0..n} Stirling1(n, m)*2^(n-m)*x^m. - Vladeta Jovovic, Feb 11 2003 T(n, m) = S1(n, m)*2^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle). The production matrix below is A038207 with the first row removed. With the initial index n = 0, the associated differential raising operator is R = e^(2D)*x = (2+x)*e^(2D) with D = d/dx, i.e., R p_n(x) = p_(n+1)(x) where p_n(x) is the n-th unsigned row polynomial and p_0(x) = 1, so p_(n+1)(x) = (2+x) * p_n(2+x). - Tom Copeland, Oct 11 2016
EXAMPLE
Triangle starts:
{1},
{2,1},
{8,6,1},
{48,44,12,1},
...
The unsigned triangle [1/(1-2x),log(1/sqrt(1-2x))] has production matrix:
2, 1,
4, 4, 1,
8, 12, 6, 1,
16, 32, 24, 8, 1,
32, 80, 80, 40, 10, 1,
64, 192, 240, 160, 60, 12, 1
which is A007318^{2} beheaded. (End)
MATHEMATICA
Table[ Rest@ CoefficientList[ Product[ z-k, {k, 0, 2p-2, 2} ], z ], {p, 6} ]
PROG
(Sage) # uses[bell_transform from A264428] # Unsigned values and an additional first column (1, 0, 0, ...).
dblfact = a.list(n)
return bell_transform(n, dblfact)
CROSSREFS
First column (unsigned triangle) is (2(n-1))!! = 1, 2, 8, 48, 384...= A000165(n-1) and the row sums (unsigned) are (2n-1)!! = 1, 3, 15, 105, 945... = A001147(n-1).
Triangle read by rows: the Bell transform of the triple factorial numbers A008544 without column 0.
+10
16
1, 2, 1, 10, 6, 1, 80, 52, 12, 1, 880, 600, 160, 20, 1, 12320, 8680, 2520, 380, 30, 1, 209440, 151200, 46480, 7840, 770, 42, 1, 4188800, 3082240, 987840, 179760, 20160, 1400, 56, 1, 96342400, 71998080, 23826880, 4583040, 562800, 45360, 2352, 72, 1
COMMENTS
Previous name was: Triangle of numbers related to triangle A048966; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497. T(n,m) = S2p(-2; n,m), a member of a sequence of triangles including S2p(-1; n,m) = A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) = A008277(n, m) (Stirling 2nd kind). T(n,1)= A008544(n-1). T(n,m), n>=m>=1, enumerates unordered n-vertex m-forests composed of m plane (aka ordered) increasing (rooted) trees where vertices of out-degree r>=0 come in r+1 different types (like an (r+1)-ary vertex). Proof from the e.g.f. of the first column Y(z) = 1 - (1-3*x)^(1/3) and the F. Bergeron et al. eq. (8) Y'(z)= phi(Y(z)), Y(0) = 0, with out-degree o.g.f. phi(w)=1/(1-w)^2. - Wolfdieter Lang, Oct 12 2007 Also the Bell transform of the triple factorial numbers A008544 which adds a first column (1,0,0 ...) on the left side of the triangle. For the definition of the Bell transform see A264428. See A051141 for the triple factorial numbers A032031 and A203412 for the triple factorial numbers A007559 as well as A039683 and A132062 for the case of double factorial numbers. - Peter Luschny, Dec 21 2015
LINKS
F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of increasing trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
FORMULA
T(n, m) = n!* A048966(n, m)/(m!*3^(n-m)); T(n+1, m) = (3*n-m)*T(n, m)+ T(n, m-1), for n >= m >= 1, with T(n, m) = 0, for n<m, and T(n, 0) = 0, T(1, 1) = 1.
E.g.f. of m-th column: ( 1 - (1-3*x)^(1/3) )^m/m!.
For a formula expressed as special values of hypergeometric functions 3F2 see the Maple program below. - Karol A. Penson, Feb 06 2004
EXAMPLE
Triangle begins:
1;
2, 1;
10, 6, 1;
80, 52, 12, 1;
880, 600, 160, 20, 1;
12320, 8680, 2520, 380, 30, 1;
209440, 151200, 46480, 7840, 770, 42, 1;
Tree combinatorics for T(3,2)=6: Consider first the unordered forest of m=2 plane trees with n=3 vertices, namely one vertex with out-degree r=0 (root) and two different trees with two vertices (one root with out-degree r=1 and a leaf with r=0). The 6 increasing labelings come then from the forest with rooted (x) trees x, o-x (1,(3,2)), (2,(3,1)) and (3,(2,1)) and similarly from the second forest x, x-o (1,(2,3)), (2,(1,3)) and (3,(1,2)).
MAPLE
T := (n, m) -> 3^n/m!*(1/3*m*GAMMA(n-1/3)*hypergeom([1-1/3*m, 2/3-1/3*m, 1/3-1/3*m], [2/3, 4/3-n], 1)/GAMMA(2/3)-1/6*m*(m-1)*GAMMA(n-2/3)*hypergeom( [1-1/3*m, 2/3-1/3*m, 4/3-1/3*m], [4/3, 5/3-n], 1)/Pi*3^(1/2)*GAMMA(2/3)):
for n from 1 to 6 do seq(simplify(T(n, k)), k=1..n) od;
# The function BellMatrix is defined in A264428. # Adds (1, 0, 0, 0, ..) as column 0.
BellMatrix(n -> mul(3*k+2, k=(0..n-1)), 9); # Peter Luschny, Jan 29 2016
MATHEMATICA
(* First program *)
T[1, 1]= 1; T[_, 0]= 0; T[0, _]= 0; T[n_, m_]:= (3*(n-1)-m)*T[n-1, m]+T[n-1, m-1];
Flatten[Table[T[n, m], {n, 12}, {m, n}] ][[1 ;; 45]] (* Jean-François Alcover, Jun 16 2011, after recurrence *) (* Second program *)
f[n_, m_]:= m/n Sum[Binomial[k, n-m-k] 3^k (-1)^(n-m-k) Binomial[n+k-1, n-1], {k, 0, n-m}]; Table[n! f[n, m]/(m! 3^(n-m)), {n, 12}, {m, n}]//Flatten (* Michael De Vlieger, Dec 23 2015 *) (* Third program *)
rows = 12;
T[n_, m_]:= BellY[n, m, Table[Product[3k+2, {k, 0, j-1}], {j, 0, rows}]];
PROG
(Sage) # uses [bell_transform from A264428] triplefactorial = lambda n: prod(3*k+2 for k in (0..n-1))
trifact = [triplefactorial(k) for k in (0..n)]
return bell_transform(n, trifact)
(Magma)
if k eq 0 then return 0;
elif k eq n then return 1;
else return (3*(n-1)-k)*T(n-1, k) + T(n-1, k-1);
end if;
end function;
[T(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2023
Generalized Stirling number triangle of first kind.
+10
12
1, -4, 1, 32, -12, 1, -384, 176, -24, 1, 6144, -3200, 560, -40, 1, -122880, 70144, -14400, 1360, -60, 1, 2949120, -1806336, 415744, -47040, 2800, -84, 1, -82575360, 53526528, -13447168, 1732864, -125440, 5152, -112, 1, 2642411520, -1795424256, 483835904
COMMENTS
a(n,m) = R_n^m(a=0, b=4) in the notation of the given 1961 and 1962 references.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 4*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference). This is the signed Stirling1 triangle with diagonal d >= 0 (main diagonal d = 0) scaled with 4^d.
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+4) ( A047053) giving unsigned values and adding 1, 0, 0, 0, ... as column 0. For the definition of the Bell transform, see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015
FORMULA
a(n, m) = a(n-1, m-1) - 4*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) := 0 for n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 4*x)/4)^m/m!.
a(n, m) = S1(n, m)*4^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
EXAMPLE
Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
1;
-4, 1;
32, -12, 1;
-384, 176, -24, 1;
6144, -3200, 560, -40, 1,
-122880, 70144, -14400, 1360, -60, 1;
...
3rd row o.g.f.: E(3,x) = 32*x - 12*x^2 + x^3.
MATHEMATICA
Table[StirlingS1[n, m] 4^(n - m), {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)
PROG
(Sage) # uses[bell_transform from A264428] # Unsigned values and an additional first column (1, 0, 0, 0, ...).
multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1))
mfact = [multifact_4_4(k) for k in (0..n)]
return bell_transform(n, mfact)
CROSSREFS
First (m=1) column sequence is: A047053(n-1). Row sums (signed triangle): A008545(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A007696(n).
Generalized Stirling number triangle of first kind.
+10
8
1, -5, 1, 50, -15, 1, -750, 275, -30, 1, 15000, -6250, 875, -50, 1, -375000, 171250, -28125, 2125, -75, 1, 11250000, -5512500, 1015000, -91875, 4375, -105, 1, -393750000, 204187500, -41037500, 4230625
COMMENTS
a(n,m) = R_n^m(a=0, b=5) in the notation of the given 1961 and 1962 references.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 5*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference). This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 5^d.
FORMULA
a(n, m) = a(n-1, m-1) - 5*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) := 0 for n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 5*x)/5)^m/m!.
a(n, m) = S1(n, m)*5^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
EXAMPLE
Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
1;
-5, 1;
50, -15, 1;
-750, 275, -30, 1;
15000, -6250, 875, -50, 1;
-375000, 171250, -28125, 2125, -75, 1;
...
3rd row o.g.f.: E(3,x) = 50*x - 15*x^2 + x^3.
CROSSREFS
First (m=1) column sequence: A052562(n-1). Row sums (signed triangle): A008546(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A008548(n).
Generalized Stirling number triangle of first kind.
+10
7
1, -6, 1, 72, -18, 1, -1296, 396, -36, 1, 31104, -10800, 1260, -60, 1, -933120, 355104, -48600, 3060, -90, 1, 33592320, -13716864, 2104704, -158760, 6300, -126, 1, -1410877440, 609700608, -102114432, 8772624, -423360, 11592, -168
COMMENTS
a(n,m) = R_n^m(a=0, b=6) in the notation of the given 1961 and 1962 references.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x-6*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference). This is the signed Stirling1 triangle A008275 with diagonal d >= 0 (main diagonal d = 0) scaled with 6^d.
FORMULA
a(n, m) = a(n-1, m-1) - 6*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 6*x)/6)^m)/m!.
a(n, m) = S1(n, m)*6^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).
EXAMPLE
Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
1;
-6, 1;
72, -18, 1;
-1296, 396, -36, 1;
31104, -10800, 1260, -60, 1;
-933120, 355104, -48600, 3060, -90, 1;
...
3rd row o.g.f.: E(3,x) = 72*x - 18*x^2 + x^3.
CROSSREFS
First (m=1) column sequence is: A047058(n-1). Row sums (signed triangle): A008543(n-1)*(-1)^(n-1). Row sums (unsigned triangle): A008542(n).
Generalized Stirling number triangle of first kind.
+10
6
1, -7, 1, 98, -21, 1, -2058, 539, -42, 1, 57624, -17150, 1715, -70, 1, -2016840, 657874, -77175, 4165, -105, 1, 84707280, -29647548, 3899224, -252105, 8575, -147, 1, -4150656720, 1537437132, -220709524, 16252369, -672280, 15778, -196, 1
COMMENTS
T(n,m) = R_n^m(a=0, b=7) in the notation of the given 1962 reference.
T(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} T(n,m)*x^m = Product_{j=0..n-1} (x-7*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference). For integers n, m >= 0 and complex numbers a, b (with b <> 0), the numbers R_n^m(a,b) were introduced by Mitrinovic (1961) and further examined by Mitrinovic and Mitrinovic (1962).
They are defined via Product_{r=0..n-1} (x - (a + b*r)) = Sum_{m=0..n} R_n^m(a,b)*x^m for n >= 0. As a result, R_n^m(a,b) = R_{n-1}^{m-1}(a,b) - (a + b*(n-1))*R_{n-1}^m(a,b) for n >= m >= 1 with R_1^0(a,b) = a, R_1^1(a,b) = 1, R_n^m(a,b) = 0 for n < m, and R_0^0(a,b) = 1.
With a = 0 and b = 1, we get the Stirling numbers of the first kind S1(n,m) = R_n^m(a=0, b=1) = A048994(n,m). We have R_n^m(a,b) = Sum_{k=0}^{n-m} (-1)^k * a^k * b^(n-m-k) * binomial(m+k, k) * S1(n, m+k) for n >= m >= 0.
For the current array, T(n,m) = R_n^m(a=0, b=7) but with no zero row or column. (End)
FORMULA
T(n, m) = T(n-1, m-1) - 7*(n-1)*T(n-1, m) for n >= m >= 1, T(n, m) = 0 for n < m, T(n, 0) = 0 for n >= 1, and T(0, 0) = 1.
Sum_{k=0..n} T(n, k) = (-1)^(n-1)* A049209(n-1). Sum_{k=0..n} (-1)^(n-k)*T(n, k) = A045754(n). E.g.f. for m-th column of signed triangle: (log(1 + 7*x)/7)^m/m!.
T(n,m) = 7^(n-m)*S1(n,m) with the (signed) Stirling1 triangle S1(n,m) = A008275(n,m). Bivariate e.g.f.-o.g.f.: Sum_{n,m >= 1} T(n,m)*x^n*y^m/n! = exp((y/7)*log(1 + 7*x)) - 1 = (1 + 7*x)^(y/7) - 1. - Petros Hadjicostas, Jun 07 2020
EXAMPLE
Triangle T(n,m) (with rows n >= 1 and columns m = 1..n) begins:
1;
-7, 1;
98, -21, 1;
-2058, 539, -42, 1;
57624, -17150, 1715, -70, 1;
-2016840, 657874, -77175, 4165, -105, 1;
...
3rd row o.g.f.: E(3,x) = Product_{j=0..2} (x - 7*j) = 98*x - 21*x^2 + x^3.
MATHEMATICA
Table[7^(n-k)*StirlingS1[n, k], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Feb 22 2022 *)
PROG
(Magma) [7^(n-k)*StirlingFirst(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 22 2022 (Sage) flatten([[(-7)^(n-k)*stirling_number1(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 22 2022
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