A255961 -id:A255961 - cp's OEIS frontend

A257673 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: row n is the inverse binomial transform of the n-th row of array A255961, which has the Euler transform of (j->j*k) in column k.

1, 0, 1, 0, 3, 1, 0, 6, 6, 1, 0, 13, 21, 9, 1, 0, 24, 62, 45, 12, 1, 0, 48, 162, 174, 78, 15, 1, 0, 86, 396, 576, 376, 120, 18, 1, 0, 160, 917, 1719, 1509, 695, 171, 21, 1, 0, 282, 2036, 4761, 5340, 3285, 1158, 231, 24, 1, 0, 500, 4380, 12441, 17234, 13473, 6309, 1792, 300, 27, 1

Offset: 0

Alois P. Heinz, May 03 2015

T is the convolution triangle of the number of plane partitions (A000219). - Peter Luschny, Oct 19 2022

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   3,    1;
  0,   6,    6,    1;
  0,  13,   21,    9,    1;
  0,  24,   62,   45,   12,    1;
  0,  48,  162,  174,   78,   15,    1;
  0,  86,  396,  576,  376,  120,   18,   1;
  0, 160,  917, 1719, 1509,  695,  171,  21,  1;
  0, 282, 2036, 4761, 5340, 3285, 1158, 231, 24, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A000219 (for n>0), A321947, A321948, A321949, A321950, A321951, A321952, A321953, A321954, A321955.
Main diagonal and lower diagonals give: A000012, A008585, A081266.
Row sums give A257674.
T(2n,n) give A257675.
Cf. A255961.

A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      A(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);
# Uses function PMatrix from A357368.
PMatrix(10, A000219); # Peter Luschny, Oct 19 2022

A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[2, j], {j, 1, n}]/n];
T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A255961(n,k-i).

G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j)^j)^k.

A055998 a(n) = n*(n+5)/2.

Original entry on oeis.org

0, 3, 7, 12, 18, 25, 33, 42, 52, 63, 75, 88, 102, 117, 133, 150, 168, 187, 207, 228, 250, 273, 297, 322, 348, 375, 403, 432, 462, 493, 525, 558, 592, 627, 663, 700, 738, 777, 817, 858, 900, 943, 987, 1032, 1078, 1125, 1173, 1222, 1272

Offset: 0

Barry E. Williams, Jun 14 2000

If X is an n-set and Y a fixed (n-3)-subset of X then a(n-3) is equal to the number of 2-subsets of X intersecting Y. - Milan Janjic, Aug 15 2007
Bisection of A165157. - Jaroslav Krizek, Sep 05 2009
a(n) is the number of (w,x,y) having all terms in {0,...,n} and w=x+y-1. - Clark Kimberling, Jun 02 2012
Numbers m >= 0 such that 8m+25 is a square. - Bruce J. Nicholson, Jul 26 2017
a(n-1) = 3*(n-1) + (n-1)*(n-2)/2 is the number of connected, loopless, non-oriented, multi-edge vertex-labeled graphs with n edges and 3 vertices. Labeled multigraph analog of A253186. There are 3*(n-1) graphs with the 3 vertices on a chain (3 ways to label the middle graph, n-1 ways to pack edges on one of connections) and binomial(n-1,2) triangular graphs (one way to label the graphs, pack 1 or 2 or ...n-2 on the 1-2 edge, ...). - R. J. Mathar, Aug 10 2017
a(n) is also the number of vertices of the quiver for PGL_{n+1} (see Shen). - Stefano Spezia, Mar 24 2020
Starting from a(2) = 7, this is the 4th column of the array: natural numbers written by antidiagonals downwards. See the illustration by Kival Ngaokrajang and the cross-references. - Andrey Zabolotskiy, Dec 21 2021

Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 10.
Milan Janjic, Two Enumerative Functions.
Kival Ngaokrajang, Illustration from A000027 (contains errors).
Linhui Shen, Duals of semisimple Poisson-Lie groups and cluster theory of moduli spaces of G-local systems, arXiv:2003.07901 [math.RT], 2020. See p. 8.
Leo Tavares, Illustration: Truncated Point Triangles.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

a(n) = A095660(n+1, 2): third column of (1, 3)-Pascal triangle.
Cf. A000096, A000217, A001477, A002522.
Row n=2 of A255961.
Cf. other rows, columns and diagonals of A000027 written as a table: A034856, A046691, A052905, A055999, A155212, A051936, A056000, A183897, A056115, A051938; A000124, A022856, A152950, A145018, A077169, A166136, A167487, A173036; A059993, A090288, A054000, A142463, A056220, A001105, A001844, A058331, A051890, A097080, A093328, A137882.

[n*(n+5)/2: n in [0..50]]; // G. C. Greubel, Apr 05 2019

f[n_]:=n*(n+5)/2; f[Range[0,50]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2011 *)

a(n)=n*(n+5)/2 \\ Charles R Greathouse IV, Sep 24 2015

[n*(n+5)/2 for n in (0..50)] # G. C. Greubel, Apr 05 2019

G.f.: x*(3-2*x)/(1-x)^3.
a(n) = A027379(n), n > 0.
a(n) = A126890(n,2) for n > 1. - Reinhard Zumkeller, Dec 30 2006
a(n) = A000217(n) + A005843(n). - Reinhard Zumkeller, Sep 24 2008
If we define f(n,i,m) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-m-j), then a(n) = -f(n,n-1,3), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = A167544(n+8). - Philippe Deléham, Nov 25 2009
a(n) = a(n-1) + n + 2 with a(0)=0. - Vincenzo Librandi, Aug 07 2010
a(n) = Sum_{k=1..n} (k+2). - Gary Detlefs, Aug 10 2010
a(n) = A034856(n+1) - 1 = A000217(n+2) - 3. - Jaroslav Krizek, Sep 05 2009
Sum_{n>=1} 1/a(n) = 137/150. - R. J. Mathar, Jul 14 2012
a(n) = 3*n + A000217(n-1) = 3*n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) = Sum_{i=3..n+2} i. - Wesley Ivan Hurt, Jun 28 2013
a(n) = 3*A000217(n) - 2*A000217(n-1). - Bruno Berselli, Dec 17 2014
a(n) = A046691(n) + 1. Also, a(n) = A052905(n-1) + 2 = A055999(n-1) + 3 for n>0. - Andrey Zabolotskiy, May 18 2016
E.g.f.: x*(6+x)*exp(x)/2. - G. C. Greubel, Apr 05 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/5 - 47/150. - Amiram Eldar, Jan 10 2021
From Amiram Eldar, Feb 12 2024: (Start)
Product_{n>=1} (1 - 1/a(n)) = -5*cos(sqrt(33)*Pi/2)/(4*Pi).
Product_{n>=1} (1 + 1/a(n)) = 15*cos(sqrt(17)*Pi/2)/(2*Pi). (End)

A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 3, 0, 1, 4, 9, 10, 5, 0, 1, 5, 14, 22, 20, 7, 0, 1, 6, 20, 40, 51, 36, 11, 0, 1, 7, 27, 65, 105, 108, 65, 15, 0, 1, 8, 35, 98, 190, 252, 221, 110, 22, 0, 1, 9, 44, 140, 315, 506, 574, 429, 185, 30, 0, 1, 10, 54, 192, 490, 918, 1265, 1240, 810, 300, 42, 0

Offset: 0

Alois P. Heinz, Sep 09 2008

A(n,k) is also the number of partitions of n into parts of k kinds.
In general, column k > 0 is asymptotic to k^((k+1)/4) * exp(Pi*sqrt(2*k*n/3)) / (2^((3*k+5)/4) * 3^((k+1)/4) * n^((k+3)/4)) * (1 - (Pi*k^(3/2)/(24*sqrt(6)) + sqrt(3)*(k+1)*(k+3)/(8*Pi*sqrt(2*k))) / sqrt(n)). - Vaclav Kotesovec, Feb 28 2015, extended Jan 16 2017
When k is a prime power greater than 1, A(n,k) is the number of conjugacy classes of n X n matrices over a field with k elements that contain an upper-triangular matrix. - Geoffrey Critzer, Nov 11 2022

			Square array begins:
  1,   1,   1,   1,   1,   1, ...
  0,   1,   2,   3,   4,   5, ...
  0,   2,   5,   9,  14,  20, ...
  0,   3,  10,  22,  40,  65, ...
  0,   5,  20,  51, 105, 190, ...
  0,   7,  36, 108, 252, 506, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-44. (Annotated scanned copy)
G. E. Bergum and V. E. Hoggatt, Jr., Numerator polynomial coefficient array for the convolved Fibonacci sequence, Fib. Quart., 14 (1976), 43-48. See Table 1.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
N. J. A. Sloane, Transforms
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Columns k=0-24 give: A000007, A000041, A000712, A000716, A023003, A023004, A023005, A023006, A023007, A023008, A023009, A023010, A005758, A023011, A023012, A023013, A023014, A023015, A023016, A023017, A023018, A023019, A023020, A023021, A006922.
Cf. A082556 (k=30), A082557 (k=32), A082558 (k=48), A082559 (k=64).
Rows n=0-4 give: A000012, A001477, A000096, A006503, A006504.
Main diagonal gives A008485.
Antidiagonal sums give A067687.
Cf. A060642, A122768, A246935, A255961, A261718.

# DedekindEta is defined in A000594.
A144064Column(k, len) = DedekindEta(len, -k)
for n in 0:8 A144064Column(n, 6) |> println end # Peter Luschny, Mar 10 2018

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: A:= (n,k)-> etr(j->k)(n): seq(seq(A(n, d-n), n=0..d), d=0..14);

a[0, ] = 1; a[, 0] = 0; a[n_, k_] := SeriesCoefficient[ Product[1/(1 - x^j)^k, {j, 1, n}], {x, 0, n}]; Table[a[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d*p[d], {d, Divisors[j]} ]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[k&][n]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)

Mat(apply( {A144064_col(k,nMax=9)=Col(1/eta('x+O('x^nMax))^k,nMax)}, [0..9])) \\ M. F. Hasler, Aug 04 2024

G.f. of column k: Product_{j>=1} 1/(1-x^j)^k.

A(n,k) = Sum_{i=0..k} binomial(k,i) * A060642(n,k-i):

A161870 Convolution square of A000219.

Original entry on oeis.org

1, 2, 7, 18, 47, 110, 258, 568, 1237, 2600, 5380, 10870, 21652, 42350, 81778, 155676, 292964, 544846, 1003078, 1828128, 3301952, 5911740, 10499385, 18502582, 32371011, 56240816, 97073055, 166497412, 283870383, 481212656, 811287037, 1360575284, 2270274785, 3769835178, 6230705170, 10251665550, 16794445441

Offset: 0

Gary W. Adamson, Jun 20 2009

nonn

Equals [1,2,3,...] * [1,0,4,0,10,0,20,...] * [1,0,0,6,0,0,21,...] * [1,0,0,0,8,0,0,0,36,...] * ... - Gary W. Adamson, Jul 06 2009

Number of pairs of planar partitions of u and v where u + v = n. - Joerg Arndt, Apr 22 2014

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
Paul Martin, Eric C. Rowell, and Fiona Torzewska, Classification of charge-conserving loop braid representations, arXiv:2301.13831 [math.QA], 2023.

Cf. A000219.

Column k=2 of A255961.

a:= proc(n) option remember; `if`(n=0, 1, 2*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 12 2015

nn = 36; CoefficientList[Series[Product[1/(1 - x^i)^(2 i), {i, 1, nn}] , {x, 0, nn}], x] (* Geoffrey Critzer, Nov 29 2014 *)

N=66;x='x+O('x^N); Vec(1/prod(k=1,N,(1-x^k)^k)^2) \\ Joerg Arndt, Apr 22 2014

G.f.: 1 / prod(k>=1, (1-x^k)^k )^2. - Joerg Arndt, Apr 22 2014
a(n) ~ Zeta(3)^(2/9) * exp(1/6 + 3*n^(2/3)*(Zeta(3)/2)^(1/3)) / (A^2 * 2^(1/18) * sqrt(3*Pi) * n^(13/18)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

Added more terms, Joerg Arndt, Apr 22 2014

A255672 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(k*n).

Original entry on oeis.org

1, 1, 7, 37, 215, 1251, 7459, 44885, 272727, 1668313, 10263057, 63423482, 393440867, 2448542136, 15280435191, 95588065737, 599213418327, 3763242239317, 23673166664695, 149138199543613, 940796936557265, 5941862248557566, 37568309060087582, 237767215209245583

Offset: 0

Vaclav Kotesovec, Mar 01 2015

nonn

Number of partitions of n when parts i are of n*i kinds. - Alois P. Heinz, Nov 23 2018
From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 3 and all positive integers n and k. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from Vaclav Kotesovec)

Cf. A008485, A252782, A270913, A270922, A380290.

Main diagonal of A255961.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

Table[SeriesCoefficient[Product[1/(1-x^k)^(k*n),{k,1,n}],{x,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 01 2015 *)

a(n) ~ c * d^n / sqrt(n), where d = 6.468409145117839606941857350154192468889057616577..., c = 0.25864792865819067933968646380369970564... . - Vaclav Kotesovec, Mar 01 2015

a(n) = [x^n] exp(n*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 30 2018

A255610 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k).

Original entry on oeis.org

1, 3, 12, 37, 111, 303, 804, 2022, 4950, 11715, 27081, 61083, 135112, 293142, 625620, 1314267, 2722323, 5564172, 11234865, 22424904, 44284545, 86573147, 167648418, 321746907, 612274678, 1155782109, 2165116416, 4026391221, 7435806048, 13641093684, 24865920932

Offset: 0

Vaclav Kotesovec, Feb 28 2015

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
Vaclav Kotesovec, Graph - The asymptotic ratio (250000 terms)
Eric Weisstein's World of Mathematics, Plane Partition
Wikipedia, Plane partition

Cf. A000219, A161870, A255611, A255612, A255613, A255614, A193427.

Column k=3 of A255961.

a:= proc(n) option remember; `if`(n=0, 1, 3*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(3*k),{k,1,nmax}],{x,0,nmax}],x]

G.f.: Product_{k>=1} 1/(1-x^k)^(3*k).
a(n) ~ Zeta(3)^(1/4) * exp(1/4 + 2^(-2/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^3 * 6^(1/4) * sqrt(Pi) * n^(3/4)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 28 2015
More precise asymptotics: a(n) ~ Zeta(3)^(1/4) * exp(1/4 + 2^(-2/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^3 * 6^(1/4) * sqrt(Pi) * n^(3/4)) * (1 - c/n^(2/3)), where c = 0.21774822... . - Vaclav Kotesovec, Oct 15 2015
G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

New name from Vaclav Kotesovec, Mar 12 2015

A299166 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k).

Original entry on oeis.org

1, 1, 2, 6, 17, 48, 132, 365, 1003, 2759, 7583, 20843, 57283, 157442, 432719, 1189317, 3268818, 8984318, 24693343, 67869557, 186539251, 512702559, 1409161449, 3873076007, 10645137706, 29258128633, 80415877302, 221022792843, 607480469466, 1669658209311, 4589050472041

Offset: 0

Ilya Gutkovskiy, Feb 04 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Antidiagonal sums of A255961.

Cf. A000219, A067687, A299105, A299106, A299108, A299162, A299164, A299167.

b:= proc(n, k) option remember; `if`(n=0, 1, k*add(
       b(n-j, k)*numtheory[sigma][2](j), j=1..n)/n)
    end:
a:= n-> add(b(n-j, j), j=0..n):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 04 2018

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

G.f.: 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k).

a(0) = 1; a(n) = Sum_{k=1..n} A000219(k-1)*a(n-k).

A193427 G.f.: Product_{k>=1} 1/(1-x^k)^(8*k).

Original entry on oeis.org

1, 8, 52, 272, 1266, 5344, 20992, 77584, 272727, 917936, 2975492, 9328736, 28391410, 84122688, 243265848, 688008048, 1906476351, 5184024112, 13851270944, 36409640400, 94255399886, 240529147072, 605574003464, 1505340071744

Offset: 0

Martin Y. Veillette, Jul 28 2011

nonn

Previous name was: Number of plane partitions of n into parts of 8 kinds.

In general, if g.f. = Product_{k>=1} 1/(1-x^k)^(m*k) and m > 0, then a(n) ~ 2^(m/36 - 1/3) * exp(m/12 + 3 * 2^(-2/3) * m^(1/3) * zeta(3)^(1/3) * n^(2/3)) * (m*zeta(3))^(m/36 + 1/6) / (A^m * sqrt(3*Pi) * n^(m/36 + 2/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.

Cf. A000219 (m=1), A161870 (m=2), A255610 (m=3), A255611 (m=4), A255612 (m=5), A255613 (m=6), A255614 (m=7).
Cf. A023007, A023003, A000712.
Column k=8 of A255961.

a:= proc(n) option remember; `if`(n=0, 1, 8*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

ANS = Block[{kmax = 50},
  Coefficient[
   Series[Product[1/(1 - x^k)^(8 k), {k, 1, kmax}], {x, 0, kmax}], x,
   Range[0, kmax]]]
(* Second program: *)
a[n_] := a[n] = If[n==0, 1, 8*Sum[a[n-j]*DivisorSigma[2, j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

Vec(prod(k=1,100\2,(1-x^k)^(-8*k),1+O(x^101))) \\ Charles R Greathouse IV, Aug 09 2011

G.f.: Product_{k>=1} (1-x^k)^(-8*k).
a(n) ~ 2^(19/18) * zeta(3)^(7/18) * exp(2/3 + 3 * 2^(1/3) * zeta(3)^(1/3) * n^(2/3)) / (A^8 * sqrt(3*Pi) * n^(8/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 28 2015
G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018
Euler transform of 8*k. - Georg Fischer, Aug 15 2020

New name from Vaclav Kotesovec, Mar 12 2015

A255611 G.f.: Product_{k>=1} 1/(1-x^k)^(4*k).

Original entry on oeis.org

1, 4, 18, 64, 215, 660, 1938, 5400, 14527, 37728, 95278, 234344, 563506, 1326796, 3066040, 6963048, 15564661, 34282360, 74486376, 159785472, 338703796, 709957616, 1472529670, 3023894672, 6151408852, 12402137024, 24792822174, 49162962280, 96737562642

Offset: 0

Vaclav Kotesovec, Feb 28 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
Eric Weisstein's World of Mathematics, Plane Partition
Wikipedia, Plane partition

Cf. A000219, A161870, A255610, A255612, A255613, A255614, A193427.

Column k=4 of A255961.

a:= proc(n) option remember; `if`(n=0, 1, 4*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(4*k),{k,1,nmax}],{x,0,nmax}],x]

G.f.: Product_{k>=1} 1/(1-x^k)^(4*k).
a(n) ~ 2^(1/3) * Zeta(3)^(5/18) * exp(1/3 + 3 * Zeta(3)^(1/3) * n^(2/3)) / (A^4 * sqrt(3*Pi) * n^(7/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 28 2015
G.f.: exp(4*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

New name from Vaclav Kotesovec, Mar 12 2015

A255612 G.f.: Product_{k>=1} 1/(1-x^k)^(5*k).

Original entry on oeis.org

1, 5, 25, 100, 370, 1251, 4005, 12150, 35400, 99365, 270353, 715025, 1844650, 4652075, 11494605, 27872056, 66428295, 155809600, 360079225, 820715820, 1846583863, 4104572975, 9019869125, 19608423750, 42193733645, 89917531549, 189863358445, 397401303850

Offset: 0

Vaclav Kotesovec, Feb 28 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
Eric Weisstein's World of Mathematics, Plane Partition
Wikipedia, Plane partition

Cf. A000219, A161870, A255610, A255611, A255613, A255614, A193427.

Column k=5 of A255961.

a:= proc(n) option remember; `if`(n=0, 1, 5*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(5*k),{k,1,nmax}],{x,0,nmax}],x]

G.f.: Product_{k>=1} 1/(1-x^k)^(5*k).
a(n) ~ 5^(11/36) * Zeta(3)^(11/36) * exp(5/12 + 3 * 2^(-2/3) * 5^(1/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^5 * 2^(7/36) * sqrt(3*Pi) * n^(29/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 28 2015
G.f.: exp(5*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

New name from Vaclav Kotesovec, Mar 12 2015

cp's OEIS Frontend

A257673 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: row n is the inverse binomial transform of the n-th row of array A255961, which has the Euler transform of (j->j*k) in column k.

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A055998 a(n) = n*(n+5)/2.

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A144064 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->k).

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Julia

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A161870 Convolution square of A000219.

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A255672 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^(k*n).

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A255610 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k).

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A299166 Expansion of 1/(1 - x*Product_{k>=1} 1/(1 - x^k)^k).

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