A028377 -id:A028377 - cp's OEIS frontend

A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

1, 1, 4, 10, 26, 59, 141, 310, 692, 1483, 3162, 6583, 13602, 27613, 55579, 110445, 217554, 424148, 820294, 1572647, 2992892, 5652954, 10605608, 19765082, 36609945, 67405569, 123412204, 224728451, 407119735, 733878402, 1316631730, 2351322765, 4180714647, 7401898452, 13051476707, 22922301583, 40105025130, 69909106888, 121427077241, 210179991927, 362583131144

Offset: 0

N. J. A. Sloane

Number of partitions of n if there are k(k+1)/2 kinds of k (k=1,2,...). E.g., a(3)=10 because we have six kinds of 3, three kinds of 2+1 because there are three kinds of 2 and 1+1+1+1. - Emeric Deutsch, Mar 23 2005
Euler transform of the triangular numbers 1,3,6,10,...
Equals A028377: [1, 1, 3, 9, 19, 46, 100, ...] convolved with the aerated version of A000294: [1, 0, 1, 0, 4, 0, 10, 0, 26, 0, 59, ...]. - Gary W. Adamson, Jun 13 2009
The formula for p3(n) in the article by S. Finch (page 2) is incomplete, terms with n^(1/2) and n^(1/4) are also needed. These terms are in the article by Mustonen and Rajesh (page 2) and agree with my results, but in both articles the multiplicative constant is marked only as C, resp. c3(m). The following is a closed form of this constant: Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8)) / (A^(1/2) * 2^(157/96) * 15^(13/96)) = A255939 = 0.213595160470..., where A = A074962 is the Glaisher-Kinkelin constant and Zeta(3) = A002117. - Vaclav Kotesovec, Mar 11 2015 [The new version of "Integer Partitions" by S. Finch contains the missing terms, see pages 2 and 5. - Vaclav Kotesovec, May 12 2015]
Number of solid partitions of corner-hook volume n (see arXiv:2009.00592 among links for definition). E.g., a(2) = 1 because there is only one solid partition [[[2]]] with cohook volume 2; a(3) = 4 because we have three solid partitions with two 1's (entry at (1,1,1) contributes 1, another entry at (2,1,1) or (1,2,1) or (1,1,2) contributes 2 to corner-hook volume) and one solid partition with single entry 3 (which contributes 3 to the corner-hook volume). Namely as 3D arrays [[[1],[1]]],[[[1]],[[1]]],[[[1]],[[1]]], [[[3]]]. - Alimzhan Amanov, Jul 13 2021

R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139.
V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alimzhan Amanov and Damir Yeliussizov, MacMahon's statistics on higher-dimensional partitions, arXiv:2009.00592 [math.CO], 2020. Mentions this sequence.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139. [Annotated scanned copy]
Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, J. Stat. Phys. 158 (2015) 950-967; arXiv:1406.5605 [cond-mat.stat-mech], 2014.
Steven Finch, Integer Partitions, September 22, 2004, page 2. [Cached copy, with permission of the author]
Vaclav Kotesovec, Graph - The asymptotic ratio
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, J. Phys. A 36 (2003), no. 24, 6651-6659; arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.
V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314. [Annotated scanned copy]
Damir Yeliussizov, Bounds on the number of higher-dimensional partitions, arXiv:2302.04799 [math.CO], 2023.

Cf. A000293, A007294, A007326, A255939, A028377.
Cf. also A000041, A000219, A000335, A000391, A000417, A000428, A255965.
Cf. also A278403 (log of o.g.f.).

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> n*(n+1)/2): seq(a(n), n=0..30);  # Alois P. Heinz, Sep 08 2008

a[0] = 1; a[n_] := a[n] = 1/(2*n)*Sum[(DivisorSigma[2, k]+DivisorSigma[3, k])*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 05 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)/2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 11 2015 *)

a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^3/k, x*O(x^n))), n)) \\ Joerg Arndt, Apr 16 2010

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n+1, 2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) = (1/(2*n))*Sum_{k=1..n} (sigma[2](k)+sigma[3](k))*a(n-k). - Vladeta Jovovic, Sep 17 2002
a(n) ~ Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(7/4)*Pi^5) + 15^(1/2) * Zeta(3) * n^(1/2) / (2^(1/2)*Pi^2) + 2^(7/4) * Pi * n^(3/4) / (3*15^(1/4))) / (A^(1/2) * 2^(157/96) * 15^(13/96) * n^(61/96)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 11 2015
G.f.: exp(Sum_{k>=1} (sigma_2(k) + sigma_3(k))*x^k/(2*k)). - Ilya Gutkovskiy, Aug 21 2018

More terms from Sascha Kurz, Aug 15 2002

A028364 Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k).

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 5, 7, 9, 14, 14, 19, 23, 28, 42, 42, 56, 66, 76, 90, 132, 132, 174, 202, 227, 255, 297, 429, 429, 561, 645, 715, 785, 869, 1001, 1430, 1430, 1859, 2123, 2333, 2529, 2739, 3003, 3432, 4862, 4862, 6292, 7150, 7810, 8398, 8986, 9646, 10504, 11934, 16796

Offset: 0

Wouter Meeussen

There are several versions of a Catalan triangle: see A009766, A008315, A028364.
The subtriangle [1], [2, 3], [5, 7, 9], ..., namely T(N,M-1), for N >= 1, M=1..N, appears as one-point function in the totally asymmetric exclusion process for the parameters alpha=1=beta. See the Derrida et al. and Liggett references given under A067323, where these triangle entries are called T_{N,N+M-1} for the given alpha and beta values. See the row reversed triangle A067323.
Consider a Dyck path as a path with steps N=(0,1) and E=(1,0) from (0,0) to (n,n) that stays weakly above y=x. T(n,m) is the number of Dyck paths of semilength n+1 where the (m+1)st north step is followed by an east step. - Lara Pudwell, Apr 12 2023

			Triangle begins
   1;
   1,  2;
   2,  3,  5;
   5,  7,  9, 14;
  14, 19, 23, 28, 42;

Alois P. Heinz, Rows n = 0..140, flattened
Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
G. Chatel and V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741.

Cf. A000108 (column 0 and main diagonal), A001700 (row sums), A065097 (T(2*n-1, n-1)), A201205 (central terms).

Cf. A067323, A009766, A039598, A039599, A028377, A028378, A028376.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      expand(b(n-1, j)*`if`(i>n, x, 1)), j=1..i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b((n+1)$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 28 2015

t[n_, k_] = Sum[CatalanNumber[n-j]*CatalanNumber[j], {j, 0, k}]; Flatten[Table[t[n, k], {n, 0, 8}, {k, 0, n}]] (* Jean-François Alcover, Jul 22 2011 *)

T(n,k) = Sum_{j>=0} A039598(k,j)*A039599(n-k,j). - Philippe Deléham, Feb 18 2004
Sum_{k>=0} T(n,k) = A001700(n). T(n,k) = A067323(n,n-k), n >= k >= 0, otherwise 0. - Philippe Deléham, May 26 2005
G.f. for column sequences m >= 0: (-(c(m,x)-1)/x+c(m,x)*c(x))/x^m with the g.f. c(x) of A000108 (Catalan) and c(m,x):=sum(C(k)*x^k,k=0..m) with C(n):=A000108(n). - Wolfdieter Lang, Mar 24 2006
G.f. for column sequences m >= 0 (without leading zeros): c(x)*Sum_{k=0..m} C(m,k)*c(x)^k with the g.f. c(x) of A000108 (Catalan) and C(n,m) is the Catalan triangle A033184(n,m). - Wolfdieter Lang, Mar 24 2006
T(n,n) = T(n,k) + T(n,n-1-k) = A000108(n+1), n > 0, k = 0..floor((n+1)/2). - Yuchun Ji, Jan 09 2019
G.f. for triangle: Sum_{n>=0, m>=0} T(n, m)*x^n*y^m = (c(x)-c(xy))/(x(1-y)c(x)) with the g.f. c(x) of A000108 (Catalan). - Lara Pudwell, Apr 12 2023

A258343 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)/6).

Original entry on oeis.org

1, 1, 4, 14, 36, 101, 260, 669, 1669, 4116, 9932, 23636, 55483, 128532, 294422, 667026, 1496232, 3324720, 7323570, 15998749, 34679966, 74622839, 159454379, 338472749, 713956569, 1496950669, 3120663129, 6469901522, 13343153563, 27379250529, 55907749171

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258342, A258344, A258345, A258346.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(binomial(i+2, 3), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 28 2018

nmax=40; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)*(k+2)/6),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) + 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 28 2018

A027999 Expansion of Product(1+q^m)^(m(m-1)/2); m=1..inf.

Original entry on oeis.org

1, 0, 1, 3, 6, 13, 24, 49, 91, 181, 334, 632, 1163, 2138, 3880, 7006, 12531, 22279, 39369, 69078, 120597, 209282, 361405, 620829, 1061687, 1807014, 3062642, 5168784, 8688820, 14549659, 24274226, 40353748, 66854518, 110391391, 181695436, 298129605, 487706902

Offset: 0

N. J. A. Sloane

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - the asymptotic ratio

Cf. A027998, A028377, A258349, A258341, A258344.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(binomial(i, 2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 03 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[Binomial[i, 2], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)

a(n) ~ 7^(1/8) / (2^(47/24) * 15^(1/8) * n^(5/8)) * exp(-2025 * Zeta(3)^3 / (98*Pi^8) - 135*(15/7)^(1/4) * Zeta(3)^2 / (28*Pi^5) * n^(1/4) - 3*sqrt(15/7) * Zeta(3) / (2*Pi^2) * sqrt(n) + 2*(7/15)^(1/4) * Pi/3 * n^(3/4)), where Zeta(3) = A002117. - Vaclav Kotesovec, May 27 2015

A258341 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)).

Original entry on oeis.org

1, 2, 7, 24, 65, 184, 487, 1254, 3145, 7706, 18480, 43490, 100692, 229472, 515802, 1144416, 2508948, 5439642, 11671859, 24801738, 52221911, 109013538, 225718717, 463769652, 945915199, 1915895576, 3854803572, 7706786958, 15314564282, 30255672820, 59440488874

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A027998, A028377, A027999, A258342, A258343, A258344, A258345, A258346, A258347.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 7^(1/8) / (2^(47/24) * 15^(1/8) * n^(5/8)) * exp(2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) + 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

Original entry on oeis.org

1, 0, 2, 6, 13, 32, 69, 160, 344, 760, 1601, 3384, 7022, 14434, 29361, 59140, 118089, 233754, 459293, 895382, 1733904, 3334914, 6374654, 12111632, 22881777, 42993244, 80362496, 149464404, 276657082, 509740278, 935046158, 1707916988, 3106810873, 5629121054

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A027998, A028377, A027999, A258341, A258342, A258343, A258345, A258346, A258348.

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 7^(1/8) / (2^(43/24) * 15^(1/8) * n^(5/8)) * exp(-2025*Zeta(3)^3 / (49*Pi^8) - 135*(15/14)^(1/4) * Zeta(3)^2 / (14*Pi^5) * n^(1/4) - 3*sqrt(15/14) * Zeta(3) / Pi^2 * sqrt(n) + 2*(14/15)^(1/4)*Pi/3 * n^(3/4)), where Zeta(3) = A002117.

A258342 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)).

Original entry on oeis.org

1, 6, 39, 224, 1131, 5412, 24411, 105078, 435048, 1740312, 6755877, 25533330, 94205738, 340064322, 1203313782, 4180514846, 14279610417, 48013553310, 159086287869, 519912616614, 1677331973910, 5345927500226, 16843574682291, 52494817082952, 161923200857711

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258343, A258344, A258345, A258346, A258350.

nmax=30; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)*(k+2)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 3^(1/5) * Zeta(5)^(1/10) / (2^(91/120) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401*Pi^16 / (1749600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - Zeta(3)^2 / (25*Zeta(5)) + (343*Pi^12/(405000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (750 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (180000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5) * Zeta(3) / (2^(3/5) * (5*Zeta(5))^(2/5))) * n^(2/5) + 7*Pi^4 / (180 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5)) * n^(3/5) + 5*3^(2/5) * (5*Zeta(5)/2)^(1/5)/4 * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

A258345 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)).

Original entry on oeis.org

1, 0, 0, 6, 24, 60, 135, 354, 972, 2684, 6990, 17802, 44627, 111582, 277329, 684164, 1671984, 4050096, 9735209, 23238480, 55120950, 129940442, 304502583, 709464798, 1643920584, 3789158988, 8690016942, 19833550266, 45056952957, 101900481462, 229462378987

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258342, A258343, A258344, A258346, A258351.

nmax=40; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)*(k-2)),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ 3^(1/5) * Zeta(5)^(1/10) / (2^(91/120) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (1749600000000*Zeta(5)^3) + 49 * Pi^8 * Zeta(3) / (2700000 * Zeta(5)^2) - Zeta(3)^2 / (25*Zeta(5)) + (-343 * Pi^12 / (405000000 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (750 * 2^(4/5) * 3^(2/5) * 5^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (180000 * 2^(3/5) * 3^(4/5) * 5^(2/5) * Zeta(5)^(7/5)) + 3^(1/5) * Zeta(3) / (2^(3/5) * (5*Zeta(5))^(2/5))) * n^(2/5) - 7*Pi^4 / (180 * 2^(2/5) * 3^(1/5) * (5*Zeta(5))^(3/5)) * n^(3/5) + 5*3^(2/5) * ((5*Zeta(5))/2)^(1/5)/4 * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

A258346 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)/6).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 20, 39, 72, 144, 280, 567, 1112, 2187, 4204, 8073, 15309, 28986, 54548, 102286, 190881, 354717, 656194, 1208712, 2217624, 4052633, 7379630, 13390098, 24215587, 43649482, 78435884, 140513905, 250988186, 447037367, 794031641, 1406585604

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258342, A258343, A258344, A258345, A258352.

nmax=50; CoefficientList[Series[Product[(1+x^k)^(k*(k-1)*(k-2)/6),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (-343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) + 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) - 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

A294102 Expansion of Product_{k>=1} (1 + x^k)^(k(3k-1)/2).

Original entry on oeis.org

1, 1, 5, 17, 44, 127, 332, 866, 2182, 5412, 13119, 31292, 73516, 170136, 388829, 877653, 1959111, 4327221, 9464856, 20511598, 44067446, 93901142, 198539477, 416696608, 868448305, 1797890682, 3698350956, 7561361750, 15369154555, 31064311255, 62449795986, 124895635385, 248538538858, 492207649241

Offset: 0

Ilya Gutkovskiy, Nov 09 2017

nonn

Weigh transform of the pentagonal numbers (A000326).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(3*n-1)/2, g(n) = -1. - Seiichi Manyama, Nov 14 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000
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 arXiv version]
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]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000326, A027998, A028377, A294836, A294837, A294838.

nmax = 33; CoefficientList[Series[Product[(1 + x^k)^(k (3 k - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (3 d - 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

G.f.: Product_{k>=1} (1 + x^k)^A000326(k).
a(n) ~ exp(-225*Zeta(3)^3 / (98*Pi^8) - 9 * 5^(5/4) * Zeta(3)^2 / (4 * 7^(5/4) * Pi^5) * n^(1/4) - (3*sqrt(5/7) * Zeta(3) / (2*Pi^2)) * sqrt(n) + (2 * (7/5)^(1/4) * Pi / 3) * n^(3/4)) * 7^(1/8) / (2^(47/24) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 10 2017
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(3*d-1)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017

cp's OEIS Frontend

A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

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A028364 Triangle T(n,m) = Sum_{k=0..m} Catalan(n-k)*Catalan(k).

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A258343 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)/6).

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A027999 Expansion of Product(1+q^m)^(m(m-1)/2); m=1..inf.

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A258341 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)).

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A258344 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)).

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A258342 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)).

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A258345 Expansion of Product_{k>=1} (1+x^k)^(k*(k-1)*(k-2)).

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A258343 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)/6).

A258342 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)).

A258345 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)).

A258346 Expansion of Product_{k>=1} (1+x^k)^(k(k-1)(k-2)/6).

A294102 Expansion of Product_{k>=1} (1 + x^k)^(k(3k-1)/2).