A025748 -id:A025748 - cp's OEIS frontend

A048966 A convolution triangle of numbers obtained from A025748.

1, 3, 1, 15, 6, 1, 90, 39, 9, 1, 594, 270, 72, 12, 1, 4158, 1953, 567, 114, 15, 1, 30294, 14580, 4482, 1008, 165, 18, 1, 227205, 111456, 35721, 8667, 1620, 225, 21, 1, 1741905, 867834, 287199, 73656, 15075, 2430, 294, 24, 1, 13586859, 6857136, 2328183, 623106, 136323, 24354, 3465, 372, 27, 1

Offset: 1

Wolfdieter Lang

A generalization of the Catalan triangle A033184.

			Triangle begins:
     1;
     3,    1;
    15,    6,    1;
    90,   39,    9,    1;
   594,  270,   72,   12,    1;
  4158, 1953,  567,  114,   15,    1;

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A034000, A049213, A049223, A049224. a(n, 1)= A025748(n), a(n, 1)= 3^(n-1)*2*A034000(n-1)/n!, n >= 2. Row sums = A025756.

a048966 n k = a048966_tabl !! (n-1) !! (k-1)
a048966_row n = a048966_tabl !! (n-1)
a048966_tabl = [1] : f 2 [1] where
   f x xs = ys : f (x + 1) ys where
     ys = map (flip div x) $ zipWith (+)
          (map (* 3) $ zipWith (*) (map (3 * (x - 1) -) [1..]) (xs ++ [0]))
          (zipWith (*) [1..] ([0] ++ xs))
-- Reinhard Zumkeller, Feb 19 2014

a[n_, m_] /; n >= m >= 1 := a[n, m] = 3*(3*(n-1) - m)*a[n-1, m]/n + m*a[n-1, m-1]/n; a[n_, m_] /; n < m := 0; a[n_, 0] = 0; a[1, 1] = 1; Table[a[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Apr 26 2011, after given formula *)

a(n, m) = 3*(3*(n-1)-m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f. for m-th column: ((1-(1-9*x)^(1/3))/3)^m.

a(n,m) = m/n * sum(k=0..n-m, binomial(k,n-m-k) * 3^k*(-1)^(n-m-k) * binomial(n+k-1,n-1)). - Vladimir Kruchinin, Feb 08 2011

A127988 Sequence determining the parity of A025748.

Original entry on oeis.org

0, 8, 32, 40, 128, 136, 160, 168, 512, 520, 544, 552, 640, 648, 672, 680, 2048, 2056, 2080, 2088, 2176, 2184, 2208, 2216, 2560, 2568, 2592, 2600, 2688, 2696, 2720, 2728, 8192, 8200, 8224, 8232, 8320, 8328, 8352, 8360, 8704, 8712, 8736

Offset: 0

Paul Barry, Feb 09 2007

Odd terms occur in A025748 at positions (0),1,2,3,(8),9,10,11,(32),33,34,35,(40),...

a(n)=8*A000695(n); a(n)=4*sum{k=0..floor(log_2(n+1)), mod(floor(n/2^k),2)2^(2k+1)};

A005130 Robbins numbers: a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!; also the number of descending plane partitions whose parts do not exceed n; also the number of n X n alternating sign matrices (ASM's).

Original entry on oeis.org

1, 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700, 31095744852375, 12611311859677500, 8639383518297652500, 9995541355448167482000, 19529076234661277104897200, 64427185703425689356896743840, 358869201916137601447486156417296

Offset: 0

N. J. A. Sloane

Also known as the Andrews-Mills-Robbins-Rumsey numbers. - N. J. A. Sloane, May 24 2013
An alternating sign matrix is a matrix of 0's, 1's and -1's such that (a) the sum of each row and column is 1; (b) the nonzero entries in each row and column alternate in sign.
a(n) is odd iff n is a Jacobsthal number (A001045) [Frey and Sellers, 2000]. - Gary W. Adamson, May 27 2009

			G.f. = 1 + x + 2*x^2 + 7*x^3 + 42*x^4 + 429*x^5 + 7436*x^6 + 218348*x^7 + ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 71, 557, 573.
D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; A_n on page 4, D_r on page 197.
C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, Chapter 75, pp. 385-386.
C. A. Pickover, Wonders of Numbers, "Princeton Numbers", Chapter 83, Oxford Univ. Press NY 2001.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
T. Amdeberhan and V. H. Moll, Arithmetic properties of plane partitions, El. J. Comb. 18 (2) (2011) # P1.
G. E. Andrews, Plane partitions (III): the Weak Macdonald Conjecture, Invent. Math., 53 (1979), 193-225. (See Theorem 10.)
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
Paul Barry, A Riordan array family for some integrable lattice models, arXiv:2409.09547 [math.CO], 2024.
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 7.
M. T. Batchelor, J. de Gier, and B. Nienhuis, The quantum symmetric XXZ chain at Delta=-1/2, alternating sign matrices and plane partitions, arXiv:cond-mat/0101385 [cond-mat.stat-mech], 2001.
Andrew Beveridge, Ian Calaway, and Kristin Heysse, de Finetti Lattices and Magog Triangles, arXiv:1912.12319 [math.CO], 2019.
E. Beyerstedt, V. H. Moll, and X. Sun, The p-adic Valuation of the ASM Numbers, J. Int. Seq. 14 (2011) # 11.8.7.
Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 33.
Sara C. Billey, Brendon Rhoades, and Vasu Tewari, Boolean product polynomials, Schur positivity, and Chern plethysm, arXiv:1902.11165 [math.CO], 2019.
D. M. Bressoud and J. Propp, How the alternating sign matrix conjecture was solved, Notices Amer. Math. Soc., 46 (No. 6, 1999), 637-646.
H. Cheballah, S. Giraudo, and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013-2015.
M. Ciucu, The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions, J. Combin. Theory Ser. A 86 (1999), 382-389.
F. Colomo and A. G. Pronko, On the refined 3-enumeration of alternating sign matrices, arXiv:math-ph/0404045, 2004; Advances in Applied Mathematics 34 (2005) 798.
F. Colomo and A. G. Pronko, Square ice, alternating sign matrices and classical orthogonal polynomials, arXiv:math-ph/0411076, 2004; JSTAT (2005) P01005.
G. Conant, Magmas and Magog Triangles, 2014.
J. de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.
P. Di Francesco, A refined Razumov-Stroganov conjecture II, arXiv:cond-mat/0409576 [cond-mat.stat-mech], 2004.
P. Di Francesco, Twenty Vertex model and domino tilings of the Aztec triangle, arXiv:2102.02920 [math.CO], 2021. Mentions this sequence.
P. Di Francesco, P. Zinn-Justin, and J.-B. Zuber, Determinant formulas for some tiling problems..., arXiv:math-ph/0410002, 2004.
FindStat - Combinatorial Statistic Finder, Alternating sign matrices
I. Fischer, The number of monotone triangles with prescribed bottom row, arXiv:math/0501102 [math.CO], 2005.
Ilse Fischer and Manjil P. Saikia, Refined Enumeration of Symmetry Classes of Alternating Sign Matrices, arXiv:1906.07723 [math.CO], 2019.
Ilse Fischer and Matjaz Konvalinka, A bijective proof of the ASM theorem, Part I: the operator formula, arXiv:1910.04198 [math.CO], 2019.
T. Fonseca and F. Balogh, The higher spin generalization of the 6-vertex model with domain wall boundary conditions and Macdonald polynomials, Journal of Algebraic Combinatorics, 2014, arXiv:1210.4527
D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, Journal of Integer Sequences Vol. 3 (2000) #00.2.3.
D. D. Frey and J. A. Sellers, Prime Power Divisors of the Number of n X n Alternating Sign Matrices
Markus Fulmek, A statistics-respecting bijection between permutation matrices and descending plane partitions without special parts, Electronic journal of combinatorics, 27(1) (2020), #P1.391.
M. Gardner, Letter to N. J. A. Sloane, Jun 20 1991.
C. Heuberger and H. Prodinger, A precise description of the p-adic valuation of the number of alternating sign matrices, Intl. J. Numb. Th. 7 (1) (2011) 57-69.
Dylan Heuer, Chelsey Morrow, Ben Noteboom, Sara Solhjem, Jessica Striker, and Corey Vorland. "Chained permutations and alternating sign matrices - Inspired by three-person chess." Discrete Mathematics 340, no. 12 (2017): 2732-2752. Also arXiv:1611.03387.
Frederick Huang, The 20 Vertex Model and Related Domino Tilings, Ph. D. Dissertation, UC Berkeley, 2023. See p. 1.
Hassan Isanloo, The volume and Ehrhart polynomial of the alternating sign matrix polytope, Cardiff University (Wales, UK 2019).
Masato Kobayashi, Weighted counting of inversions on alternating sign matrices, arXiv:1904.02265 [math.CO], 2019.
G. Kuperberg, Another proof of the alternating-sign matrix conjecture, arXiv:math/9712207 [math.CO], 1997; Internat. Math. Res. Notices, No. 3, (1996), 139-150.
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001; Ann. Math. 156 (3) (2002) 835-866
W. H. Mills, David P Robbins, and Howard Rumsey Jr., Alternating sign matrices and descending plane partitions J. Combin. Theory Ser. A 34 (1983), no. 3, 340--359. MR0700040 (85b:05013).
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, Chapter 75, pp. 385-386. [Annotated scanned copy]
J. Propp, The many faces of alternating-sign matrices, Discrete Mathematics and Theoretical Computer Science Proceedings AA (DM-CCG), 2001, 43-58.
A. V. Razumov and Yu. G. Stroganov, Spin chains and combinatorics, arXiv:cond-mat/0012141 [cond-mat.stat-mech], 2000.
Lukas Riegler, Simple enumeration formulas related to Alternating Sign Monotone Triangles and standard Young tableaux, Dissertation, Universitat Wien, 2014.
D. P. Robbins, The story of 1, 2, 7, 42, 429, 7436, ..., Math. Intellig., 13 (No. 2, 1991), 12-19.
D. P. Robbins, Symmetry classes of alternating sign matrices, arXiv:math/0008045 [math.CO], 2000.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986.
R. P. Stanley, A baker's dozen of conjectures concerning plane partitions, pp. 285-293 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, 1986. Preprint. [Annotated scanned copy]
Yu. G. Stroganov, 3-enumerated alternating sign matrices, arXiv:math-ph/0304004, 2003.
X. Sun and V. H. Moll, The p-adic Valuations of Sequences Counting Alternating Sign Matrices, JIS 12 (2009) 09.3.8.
Eric Weisstein's World of Mathematics, Alternating Sign Matrix
Eric Weisstein's World of Mathematics, Descending Plane Partition
D. Zeilberger, Proof of the alternating-sign matrix conjecture, arXiv:math/9407211 [math.CO], 1994.
D. Zeilberger, Proof of the alternating-sign matrix conjecture, Elec. J. Combin., Vol. 3 (Number 2) (1996), #R13.
D. Zeilberger, Proof of the Refined Alternating Sign Matrix Conjecture, arXiv:math/9606224 [math.CO], 1996.
D. Zeilberger, A constant term identity featuring the ubiquitous (and mysterious) Andrews-Mills-Robbins-Ramsey numbers 1,2,7,42,429,..., J. Combin. Theory, A 66 (1994), 17-27. The link is to a comment on Doron Zeilberger's home page. A backup copy is here [pdf file only, no active links]
D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954. The link is to a version on Doron Zeilberger's home page. A backup copy is here [pdf file only, no active links]
Paul Zinn-Justin, Integrability and combinatorics, arXiv:2404.13221 [math.CO], 2024. See p. 12.
Index entries for sequences related to factorial numbers
Index entries for "core" sequences

Cf. A006366, A048601, also A003827, A005156, A005158, A005160-A005164, A050204, A049503, A194827, A227833.

a:=List([0..18],n->Product([0..n-1],k->Factorial(3*k+1)/Factorial(n+k)));; Print(a); # Muniru A Asiru, Jan 02 2019

A005130 := proc(n) local k; mul((3*k+1)!/(n+k)!,k=0..n-1); end;
# Bill Gosper's approximation (for n>0):
a_prox := n -> (2^(5/12-2*n^2)*3^(-7/36+1/2*(3*n^2))*exp(1/3*Zeta(1,-1))*Pi^(1/3)) /(n^(5/36)*GAMMA(1/3)^(2/3)); # Peter Luschny, Aug 14 2014

f[n_] := Product[(3k + 1)!/(n + k)!, {k, 0, n - 1}]; Table[ f[n], {n, 0, 17}] (* Robert G. Wilson v, Jul 15 2004 *)
a[ n_] := If[ n < 0, 0, Product[(3 k + 1)! / (n + k)!, {k, 0, n - 1}]]; (* Michael Somos, May 06 2015 *)

{a(n) = if( n<0, 0, prod(k=0, n-1, (3*k + 1)! / (n + k)!))}; /* Michael Somos, Aug 30 2003 */

{a(n) = my(A); if( n<0, 0, A = Vec( (1 - (1 - 9*x + O(x^(2*n)))^(1/3)) / (3*x)); matdet( matrix(n, n, i, j, A[i+j-1])) / 3^binomial(n,2))}; /* Michael Somos, Aug 30 2003 */

from math import prod, factorial
def A005130(n): return prod(factorial(3*k+1) for k in range(n))//prod(factorial(n+k) for k in range(n)) # Chai Wah Wu, Feb 02 2022

a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!.
The Hankel transform of A025748 is a(n) * 3^binomial(n, 2). - Michael Somos, Aug 30 2003
a(n) = sqrt(A049503).
From Bill Gosper, Mar 11 2014: (Start)
A "Stirling's formula" for this sequence is
a(n) ~ 3^(5/36+(3/2)*n^2)/(2^(1/4+2*n^2)*n^(5/36))*(exp(zeta'(-1))*gamma(2/3)^2/Pi)^(1/3).
which gives results which are very close to the true values:
1.0063254118710128, 2.003523267231662,
7.0056223910285915, 42.01915917750558,
429.12582410098327, 7437.518404899576,
218380.8077275304, 1.085146545456063*^7,
9.119184824937415*^8
(End)
a(n+1) = a(n) * n! * (3*n+1)! / ((2*n)! * (2*n+1)!). - Reinhard Zumkeller, Sep 30 2014; corrected by Eric W. Weisstein, Nov 08 2016
For n>0, a(n) = 3^(n - 1/3) * BarnesG(n+1) * BarnesG(3*n)^(1/3) * Gamma(n)^(1/3) * Gamma(n + 1/3)^(2/3) / (BarnesG(2*n+1) * Gamma(1/3)^(2/3)). - Vaclav Kotesovec, Mar 04 2021

A034000 One half of triple factorial numbers.

Original entry on oeis.org

1, 5, 40, 440, 6160, 104720, 2094400, 48171200, 1252451200, 36321084800, 1162274713600, 40679614976000, 1545825369088000, 63378840132608000, 2788668965834752000, 131067441394233344000, 6553372069711667200000, 347328719694718361600000, 19450408302904228249600000

Offset: 1

Wolfdieter Lang

Preface the series with a 1, then the next term = (1, 4, 7, 10, ...) dot (1, 1, 5, 40, ...). E.g., a(5) = 6160 = (1, 4, 7, 10, 13) dot (1, 1, 5, 40, 440) = (1 + 4 + 35 + 400 + 5720). - Gary W. Adamson, May 17 2010

In other words, a(n) = Sum_{i=0..n-1} b(i)*A016777(i) where b(0)=1 and b(n)=a(n). - Michel Marcus, Dec 18 2022

G. C. Greubel, Table of n, a(n) for n = 1..375
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.

Cf. A007559, A034001, A025748, A034724, A016777.

a:=[1];; for n in [2..20] do a[n]:=(3*n-1)*a[n-1]; od; a; # G. C. Greubel, Aug 15 2019

[n le 1 select 1 else (3*n-1)*Self(n-1): n in [1..20]]; // G. C. Greubel, Aug 15 2019

A034000:=n->`if`(n=1, 1, (3*n-1)*A034000(n-1)); seq(A034000(n), n=1..20); # G. C. Greubel, Aug 15 2019

nxt[{n_,a_}]:={n+1,(3(n+1)-1)*a}; Transpose[NestList[nxt,{1,1},20]][[2]] (* Harvey P. Dale, Aug 22 2015 *)
Table[3^(n-1)*Pochhammer[5/3, n-1], {n,20}] (* G. C. Greubel, Aug 15 2019 *)

m=20; v=concat([1], vector(m-1)); for(n=2, m, v[n]=(3*n-1)*v[n-1]); v \\ G. C. Greubel, Aug 15 2019

def a(n):
    if n==1: return 1
    else: return (3*n-1)*a(n-1)
[a(n) for n in (1..20)] # G. C. Greubel, Aug 15 2019

a(n) = A007661(3n-1)/2 = A008544(n)/2.
2*a(n+1) = (3*n+2)!!! = Product_{j=0..n} (3*j+2), n >= 0.
E.g.f.: (-1 + (1-3*x)^(-2/3))/2.
a(n) = (3*n-1)!/(2*3^(n-1)*(n-1)!*A007559(n)).
a(n) ~ 3/2*2^(1/2)*Pi^(1/2)*Gamma(2/3)^-1*n^(7/6)*3^n*e^-n*n^n*{1 + 23/36*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
a(n) = 3^n*(n+2/3)!/(2/3)!, with offset 0. - Paul Barry, Sep 04 2005
D-finite with recurrence a(n) + (1-3*n)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
Sum_{n>=1} 1/a(n) = 2*(e/3)^(1/3)*(Gamma(2/3) - Gamma(2/3, 1/3)). - Amiram Eldar, Dec 18 2022

A097188 G.f. A(x) satisfies A057083(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097186.

Original entry on oeis.org

1, 3, 15, 90, 594, 4158, 30294, 227205, 1741905, 13586859, 107459703, 859677624, 6943550040, 56540336040, 463630755528, 3824953733106, 31724616256938, 264371802141150, 2212374554760150, 18583946259985260, 156636118477018620

Offset: 0

Paul D. Hanna, Aug 03 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4, eq.(23) for l=4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Cf. A004988, A025748, A034164, A057083, A097186.

Essentially identical to A025748.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1 - (1-9*x)^(1/3))/(3*x) )); // G. C. Greubel, Sep 17 2019

seq(coeff(series((1-(1-9*x)^(1/3))/(3*x), x, n+2), x, n), n = 0..25); # G. C. Greubel, Sep 17 2019

Table[FullSimplify[9^n * Gamma[n+2/3] / ((n+1) * Gamma[2/3] * Gamma[n+1])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1-(1 - 9 x)^(1/3))/(3 x), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 10 2014 *)

a(n)=polcoeff((1-(1-9*x+x^2*O(x^n))^(1/3))/(3*x),n,x)

def A097188_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1 - (1-9*x)^(1/3))/(3*x)).list()
A097188_list(25) # G. C. Greubel, Sep 17 2019

G.f.: A(x) = (1 - (1-9*x)^(1/3))/(3*x).
G.f.: A(x) = (1/x)*(series reversion of x/A057083(x)).
a(n) = A004988(n)/(n+1).
a(n) = A025748(n+1).
a(n) = 3*A034164(n-1) for n>=1.
x*A(x) is the compositional inverse of x-3*x^2+3*x^3. - Ira M. Gessel, Feb 18 2012
a(n) = 1/(n+1) * Sum_{k=1..n} binomial(k,n-k) * 3^(k)*(-1)^(n-k) * binomial(n+k,n), if n>0; a(0)=1. - Vladimir Kruchinin, Feb 07 2011
Conjecture: (n+1)*a(n) +3*(-3*n+1)*a(n-1)=0. - R. J. Mathar, Nov 16 2012
a(n) = 9^n * Gamma(n+2/3) / ((n+1) * Gamma(2/3) * Gamma(n+1)). - Vaclav Kotesovec, Feb 09 2014
Sum_{n>=0} 1/a(n) = 21/16 + 3*sqrt(3)*Pi/64 - 9*log(3)/64. - Amiram Eldar, Dec 02 2022

A034171 Related to triple factorial numbers A007559(n+1).

Original entry on oeis.org

1, 6, 42, 315, 2457, 19656, 160056, 1320462, 11003850, 92432340, 781473420, 6642524070, 56716936290, 486145168200, 4180848446520, 36059817851235, 311811366125385, 2702365173086670, 23467908082068450, 204170800313995515, 1779202688450532345, 15527587099204645920

Offset: 0

Wolfdieter Lang

Working with an offset of 1, we conjecture a(p*n) = a(n) (mod p^2) for prime p = 1 (mod 3) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/3 and 1 <= k <= (p-1)/3. Cf. A298799, A004981 and A004982. - Peter Bala, Dec 23 2019

Michael De Vlieger, Table of n, a(n) for n = 0..1050
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A007559, A025748, A034164, A035529.

Cf. A298799, A004981, A004982, A004987, A073005.

CoefficientList[Series[(-1 + (1 - 9 x)^(-1/3))/(3 x), {x, 0, 19}], x] (* Michael De Vlieger, Oct 13 2019 *)

a(n) = 3^n*A007559(n+1)/(n+1)! where A007559(n+1)=(3*n+1)!!!.
G.f.: (-1+(1-9*x)^(-1/3))/(3*x).
a(n) = A035529(n+1, 1) (first column of triangle).
Convolution of A004987(n) with A025748(n+1), n >= 0.
From R. J. Mathar, Jan 28 2020: (Start)
D-finite with recurrence: (n+1)*a(n) + 3*(-3*n-1)*a(n-1) = 0.
G.f.: (1F0(1/3;;9*x)-1)/(3*x). (End)
Sum_{n>=0} 1/a(n) = 3/8 + 3*sqrt(3)*Pi/32 + 9*log(3)/32. - Amiram Eldar, Dec 22 2022
a(n) ~ 3^(2*n+1) * n^(-2/3) / Gamma(1/3). - Amiram Eldar, Aug 19 2025

A004989 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k - 2).

Original entry on oeis.org

1, -6, -9, -36, -189, -1134, -7371, -50544, -360126, -2640924, -19806930, -151252920, -1172210130, -9197341020, -72921775230, -583374201840, -4703454502335, -38180983607190, -311811366125385, -2560135427134740, -21121117273861605, -175003543126281870, -1455711290550435555

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004988, A004990, A004991, A004992, A025748, A155579, A185047.

List([0..25], n-> 3^n*Product([0..n-1], k-> 3*k-2)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

[1] cat [3^n*(&*[3*k-2: k in [0..n-1]])/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019

a:= n-> (3^n/n!)*product(3*k-2, k=0..n-1); seq(a(n), n=0..25); # G. C. Greubel, Aug 22 2019

Table[9^n*Pochhammer[-2/3, n]/n!, {n,0,25}] (* G. C. Greubel, Aug 22 2019 *)

a(n)=if(n<0,0,prod(k=0,n-1,3*k-2)*3^n/n!)

[9^n*rising_factorial(-2/3, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019

a(n) ~ -(2/3)*Gamma(1/3)^-1*n^(-5/3)*3^(2*n)*(1 + (5/9)*n^-1 + ...).
G.f.: (1-9*x)^(2/3).
D-finite with recurrence: n*a(n) +3*(-3*n+5)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 99/128 - 5*sqrt(3)*Pi/512 - 15*log(3)/512. - Amiram Eldar, Dec 02 2022
From Peter Bala, Oct 14 2024: (Start)
a(n) = -1/(sqrt(3)*Pi) * 9^n * Gamma(2/3)*Gamma(n-2/3)/Gamma(n+1).
For n >= 1, a(n) = - Integral_{x = 0..9} x^n * w(x) dx, where w(x) = sqrt(3)/(2*Pi) * x^(1/3)*(9 - x)^(2/3)/x^2. (End)

A034164 Related to triple factorial numbers 2*A034000(n+1).

Original entry on oeis.org

1, 5, 30, 198, 1386, 10098, 75735, 580635, 4528953, 35819901, 286559208, 2314516680, 18846778680, 154543585176, 1274984577702, 10574872085646, 88123934047050, 737458184920050, 6194648753328420, 52212039492339540, 441429061162507020, 3742550735942994300

Offset: 0

Wolfdieter Lang

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A004990, A025748, A034000, A073006, A185047.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1 -3*x -(1-9*x)^(1/3))/(3*x)^2 )); // G. C. Greubel, Sep 17 2019

seq(coeff(series((1-3*x-(1-9*x)^(1/3))/(3*x)^2, x, n+2), x, n), n = 0..32); # G. C. Greubel, Sep 17 2019

CoefficientList[Series[ HypergeometricPFQ[{1, 5/3}, {3}, 9 x], {x, 0, 20}], x]
Table[FullSimplify[3^(2*n+1) * Gamma[n+5/3] / ((n+2) * Gamma[2/3] * Gamma[n+2])],{n,0,20}] (* Vaclav Kotesovec, Feb 09 2014 *)
CoefficientList[Series[(1 -3x -(1-9 x)^(1/3))/(3 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Feb 10 2014 *)

my(x='x+O('x^30)); Vec((1 -3*x -(1-9*x)^(1/3))/(3*x)^2) \\ G. C. Greubel, Sep 17 2019

def A034164_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P((1 -3*x -(1-9*x)^(1/3))/(3*x)^2).list()
A034164_list(30) # G. C. Greubel, Sep 17 2019

a(n) = 3^n*(3*n+2)!!!/(n+2)!, where (3*n+2)!!! = 2*A034000(n+1).
G.f.: (1 - 3*x - (1-9*x)^(1/3))/(3*x)^2.
G.f.: 2F1( (1, 5/3); 3; 9 x ). - Olivier Gérard, Feb 15 2011
D-finite with recurrence: (n+2)*a(n) - 3*(3*n+2)*a(n-1) = 0. - R. J. Mathar, Oct 29 2012
a(n) = 3^(2*n+1) * Gamma(n+5/3) / ((n+2) * Gamma(2/3) * Gamma(n+2)). - Vaclav Kotesovec, Feb 09 2014
Integral representation as the n-th moment of a positive function on (0,9): a(n) = Integral_{x=0..9} x^n*W(x) dx, n >= 0, where W(x) = (1/18)*9^(1/3)*sqrt(3)*x^(2/3)*(1-x/9)^(1/3)/Pi. This representation is unique as W(x) is the solution of the Hausdorff moment problem. - Karol A. Penson, Nov 07 2015
Sum_{n>=0} 1/a(n) = 15/16 + (27/64)*(Pi*sqrt(3)/3 - log(3)). - Amiram Eldar, Dec 02 2022
a(n) ~ 3^(2*n+1) * n^(-4/3) / Gamma(2/3). - Amiram Eldar, Aug 19 2025

A254287 Expansion of (1 - (1 - 3125x)^(1/5)) / (625x).

Original entry on oeis.org

1, 1250, 2343750, 5126953125, 12176513671875, 30441284179687500, 78821182250976562500, 209368765354156494140625, 567040406167507171630859375, 1559361116960644721984863281250, 4341403109719976782798767089843750, 12210196246087434701621532440185546875

Offset: 0

Vaclav Kotesovec, Jan 27 2015

In general, if k > 1 and g.f. = (1 - (1 - k^k * x)^(1/k)) / (k^(k-1) * x), then a(n) ~ k^(k*n) / (Gamma((k-1)/k) * n^((k+1)/k)).

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000108 (k=2), A254282 (k=3), A254286 (k=4).

Cf. A008546, A025748.

[Round(5^(5*n)*Gamma(n+4/5)/(Gamma(4/5)*Gamma(n+2))): n in [0..30]]; // G. C. Greubel, Aug 10 2022

CoefficientList[Series[(1-(1-3125*x)^(1/5)) / (625*x),{x,0,20}],x]
CoefficientList[Series[Hypergeometric1F1[4/5,2,3125*x],{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

[5^(5*n)*rising_factorial(4/5, n)/factorial(n+1) for n in (0..30)] # G. C. Greubel, Aug 10 2022

G.f.: (1 - (1 - 3125*x)^(1/5)) / (625*x).
a(n) ~ 3125^n / (Gamma(4/5) * n^(6/5)).
Recurrence: (n+1)*a(n) = 625*(5*n-1)*a(n-1).
a(n) = 5^(5*n) * Gamma(n+4/5) / (Gamma(4/5) * Gamma(n+2)).
E.g.f.: hypergeom([4/5], [2], 3125*x). - Vaclav Kotesovec, Jan 28 2015
From Peter Bala, Sep 01 2017: (Start)
a(n) = (-1)^n*binomial(1/5, n+1)*5^(5*n+1). Cf. A000108(n) = (-1)^n*binomial(1/2, n+1)*2^(2*n+1).
a(n) = 125^n*A025748(n+1). (End)

A059486 3-enumeration of 2n+1 X 2n+1 vertically symmetric alternating-sign matrices.

Original entry on oeis.org

1, 1, 5, 126, 16038, 10320453, 33590259846, 553104735325740, 46084184498427053436, 19430969437346561065941390, 41463730793298298041665385308325, 447814224393522724673729884056814834500, 24479424309393636290695101063892553945412075000

Offset: 0

N. J. A. Sloane, Feb 04 2001

Harry J. Smith, Table of n, a(n) for n = 0..53
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001. [Th. 3, but the formula there is incorrect]
J. Propp, The many faces of alternating-sign matrices, Discrete Mathematics and Theoretical Computer Science Proceedings AA (DM-CCG), 2001, 43-58.

Cf. A025748, A227379.

A059486 := proc(n) local i, j, t1; t1 := 3^(2*n^2)/2^(2*n^2 + n); for i to 2*n + 1 do for j to 2*n + 1 do if i mod 2 <> 0 and j mod 2 = 0 then t1 := t1*(3*j - 3*i + 1)/(3*j - 3*i) end if end do end do; t1 end proc;
e(n)= { local(A); A=Vec((1 - (1 - 9*x + O(x^(2*n + 1)))^(1/3))/(3*x)); matdet(matrix(n, n, i, j, A[i+j]))/3^n; } { for (n = 0, 100, a=e(n); if (a > 10^(10^3 - 6), break); write("b059486.txt", n, " ", a); ) } # Harry J. Smith, Jun 27 2009

a[n_] := Module[{i, j, t1}, t1 = 3^(2*n^2)/2^(2*n^2 + n); For[i = 1, i <= 2*n + 1, i++, For[j = 1, j <= 2*n + 1, j++, If[Mod[i, 2] != 0 && Mod[j, 2] == 0, t1 = t1*(3*j - 3*i + 1)/(3*j - 3*i)]]]; t1];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 23 2017, translated from Maple *)
Table[3^(2*n^2)/2^(2*n^2 + n) * Product[(2 + 6*i - 6*j)/(3 + 6*i - 6*j), {i, 0, n}, {j, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Feb 24 2019 *)

a(n)=local(A); if(n<0,0,A=Vec((1-(1-9*x+O(x^(2*n+1)))^(1/3))/(3*x)); matdet(matrix(n,n,i,j,A[i+j]))/3^n)

e(n)= { local(A); A=Vec((1 - (1 - 9*x + O(x^(2*n + 1)))^(1/3))/(3*x)); matdet(matrix(n, n, i, j, A[i+j]))/3^n; } { for (n = 0, 100, a=e(n); if (a > 10^(10^3 - 6), break); write("b059486.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 27 2009

a(n) ~ exp(1/36) * Gamma(1/3)^(1/3) * 3^(n*(4*n + 1)/2 + 11/36) * n^(1/36) / (2^(2*n*(n+1) + 7/12) * A^(1/3) * Pi^(1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Feb 24 2019

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A048966 A convolution triangle of numbers obtained from A025748.

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A127988 Sequence determining the parity of A025748.

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A005130 Robbins numbers: a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!; also the number of descending plane partitions whose parts do not exceed n; also the number of n X n alternating sign matrices (ASM's).

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A034000 One half of triple factorial numbers.

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A097188 G.f. A(x) satisfies A057083(x*A(x)) = A(x) and so equals the ratio of the g.f.s of any two adjacent diagonals of triangle A097186.

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A034171 Related to triple factorial numbers A007559(n+1).

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A004989 a(n) = (3^n/n!) * Product_{k=0..n-1} (3*k - 2).

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A254287 Expansion of (1 - (1 - 3125x)^(1/5)) / (625x).