A000335 -id:A000335 - cp's OEIS frontend

A278767 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k-1)).

1, 1, 7, 22, 71, 206, 616, 1712, 4743, 12677, 33407, 86085, 218677, 546060, 1345840, 3271893, 7861239, 18670881, 43883904, 102112483, 235401947, 537869136, 1218743007, 2739566083, 6111766043, 13536683750, 29775945929, 65065819486, 141285315728, 304935221675, 654318376244, 1396166024244, 2963068779402

Offset: 0

Ilya Gutkovskiy, Nov 28 2016

nonn

Euler transform of the hexagonal numbers (A000384).

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, Hexagonal Number
Index to sequences related to polygonal numbers

Cf. A000294, A000384, A000335, A023871.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(2*d-1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(2*k-1)).

a(n) ~ exp(-Zeta'(-1) - Zeta(3)/(2*Pi^2) - 75*Zeta(3)^3/(4*Pi^8) - 15^(5/4)*Zeta(3)^2/(2^(9/4)*Pi^5) * n^(1/4) - sqrt(15/2)*Zeta(3)/Pi^2 * sqrt(n) + 2^(9/4)*Pi/(3^(5/4)*5^(1/4)) * n^(3/4)) / (2^(67/48) * 15^(5/48) * Pi^(1/12) * n^(29/48)). - Vaclav Kotesovec, Dec 02 2016

A278769 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k-3)/2).

Original entry on oeis.org

1, 1, 8, 26, 88, 269, 843, 2456, 7115, 19892, 54756, 147355, 390517, 1017091, 2612670, 6617641, 16556913, 40933339, 100104289, 242276236, 580718077, 1379161494, 3247074738, 7581837910, 17564867853, 40388447308, 92206496318, 209069338580, 470944571003, 1054178579266, 2345477963043, 5188246121144, 11412352653001

Offset: 0

Ilya Gutkovskiy, Nov 28 2016

nonn

Euler transform of the heptagonal numbers (A000566).

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, Heptagonal Number
Index to sequences related to polygonal numbers

Cf. A000294, A000566, A000335, A023871.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(5*d-3)/2, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*(5*k-3)/2).

a(n) ~ exp(-3*Zeta'(-1)/2 - 5*Zeta(3)/(8*Pi^2) - 81*Zeta(3)^3/(2*Pi^8) - 3^(13/4)*Zeta(3)^2/(2^(7/4)*Pi^5) * n^(1/4) - 3^(3/2)*Zeta(3)/(sqrt(2)*Pi^2) * sqrt(n) + 2^(7/4)*Pi/3^(5/4) * n^(3/4)) / (2^(51/32) * 3^(3/32) * Pi^(1/8) * n^(19/32)). - Vaclav Kotesovec, Dec 02 2016

A295179 Expansion of Product_{k>=1} 1/(1 - x^k)^(3k(k-1)/2+1).

Original entry on oeis.org

1, 1, 5, 15, 44, 115, 312, 790, 2004, 4908, 11885, 28170, 65987, 152079, 346560, 779808, 1736460, 3825995, 8351733, 18064545, 38747740, 82443251, 174096564, 364991008, 759989218, 1572126699, 3231929735, 6604498620, 13419469596, 27117216441, 54508611399, 109013531864, 216956853105

Offset: 0

Ilya Gutkovskiy, Nov 16 2017

nonn

Euler transform of the centered triangular numbers (A005448).

Vaclav Kotesovec, Table of n, a(n) for n = 0..4000
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, Centered Triangular Number
Index entries for sequences related to centered polygonal numbers

Cf. A000294, A000335, A005448, A295180.

nmax = 32; CoefficientList[Series[Product[1/(1 - x^k)^(3 k (k - 1)/2 + 1), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (3 d (d - 1)/2 + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A005448(k).

a(n) ~ exp(-3*Zeta'(-1)/2 + 7*Zeta(3) / (8*Pi^2) - 225*Zeta(3)^3 / (2*Pi^8) + (Pi / (3*2^(3/4)) - 45*Zeta(3)^2 / (2^(7/4) * Pi^5)) * (5*n)^(1/4) - (3*sqrt(5/2) * Zeta(3) / Pi^2) * sqrt(n) + (2^(7/4)*Pi / (3*5^(1/4))) * n^(3/4)) / (2^(71/32) * 5^(7/32) * Pi^(1/8) * n^(23/32)). - Vaclav Kotesovec, Nov 16 2017

A292387 Expansion of Product_{k>=1} (1 - x^k)^(k(k+1)(k+2)/6).

Original entry on oeis.org

1, -1, -4, -6, -4, 19, 60, 131, 149, -4, -572, -1764, -3485, -4716, -2658, 7606, 32944, 77152, 132586, 161275, 75150, -281687, -1111029, -2560293, -4470415, -5922117, -4603551, 3799070, 25573251, 67259095, 130430051, 201158707, 232853019, 124749892, -295134275, -1260897993, -2995361708, -5515840117

Offset: 0

Ilya Gutkovskiy, Sep 15 2017

sign

Convolution inverse of A000335 (Euler transform of the tetrahedral numbers).

Cf. A000335, A258343, A292386.

nmax = 37; CoefficientList[Series[Product[(1 - x^k)^(k (k + 1) (k + 2)/6), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k)^(k*(k+1)*(k+2)/6).

A293551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 141, 86, 22, 1, 1, 1, 9, 36, 105, 216, 331, 310, 160, 30, 1, 1, 1, 10, 45, 148, 357, 672, 855, 692, 282, 42, 1, 1, 1, 11, 55, 201, 554, 1232, 1982, 2214, 1483, 500, 56, 1

Offset: 0

Ilya Gutkovskiy, Oct 11 2017

A(n,k) is the Euler transform of j -> binomial(j+k-2,k-1) evaluated at n.

			Square array begins:
1,  1,   1,   1,    1,    1,  ...
1,  1,   1,   1,    1,    1,  ...
1,  2,   3,   4,    5,    6,  ...
1,  3,   6,  10,   15,   21,  ...
1,  5,  13,  26,   45,   71,  ...
1,  7,  24,  59,  120,  216,  ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
S. Balakrishnan, S. Govindarajan, and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
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]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
N. J. A. Sloane, Transforms

Columns k=0..8 give A000012, A000041, A000219, A000294, A000335, A000391, A000417, A000428, A255965.
Main diagonal gives A293554.
Cf. A007318, A096751 (a similar but different sequence).

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*
      binomial(d+k-2, k-1), d=divisors(j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Oct 17 2017

Table[Function[k, SeriesCoefficient[E^(Sum[x^i/(i (1 - x^i)^k), {i, 1, n}]), {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

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

For asymptotics of column k see comment from Vaclav Kotesovec in A255965.

A190905 Euler transform of the swinging factorial A056040.

Original entry on oeis.org

1, 1, 3, 9, 18, 60, 117, 371, 747, 2199, 4697, 12735, 28571, 72815, 169176, 412440, 978086, 2316754, 5547293, 12909723, 30966639, 71357601, 170636159, 391242623, 930120982, 2128073530, 5023630830, 11486060090, 26918052717, 61539213693, 143227189518

Offset: 0

Peter Luschny, Jul 06 2011

nonn

M. Bernstein and N. J. A. Sloane, Some Canonical Integer Sequences, (arXiv:0205301v1), May 28 2002. [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]

Cf. A107895.

EulerTrans := proc(p) local b; b := proc(n) option remember; local d, j;
`if`(n=0,1,add(add(d*p(d),d=numtheory[divisors](j))*b(n-j),j=1..n)/n) end end:
A190905 := EulerTrans(n->n!/iquo(n,2)!^2): seq( A190905(n),n=0..30); # After Alois P. Heinz, A000335.

sf[n_] := With[{f = Floor[n/2]}, Pochhammer[f+1, n-f]/f!]; EulerTrans[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 = EulerTrans[sf]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 29 2013, after Maple *)

A302449 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(4k^2-1)/3).

Original entry on oeis.org

1, 1, 11, 46, 185, 700, 2676, 9646, 34166, 117500, 396506, 1310527, 4258313, 13607309, 42846151, 133039791, 407833188, 1235202869, 3699140386, 10960888382, 32154531807, 93437164720, 269087234273, 768340525743, 2176098269286, 6115444177489, 17058887661133

Offset: 0

Ilya Gutkovskiy, Apr 08 2018

nonn

Euler transform of A000447.

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

Cf. A000335, A000447, A023871, A279215.

nmax = 26; CoefficientList[Series[Product[1/(1 - x^k)^(k (4 k^2 - 1)/3), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (4 d^2 - 1)/3, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 26}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A000447(k).

a(n) ~ exp(5 * Zeta(5)^(1/5) * n^(4/5)/2 - Zeta(3) * n^(2/5) / (12 * Zeta(5)^(2/5)) + 4*Zeta'(-3)/3 - 1/36 - Zeta(3)^2 / (720*Zeta(5))) * A^(1/3) * Zeta(5)^(83/900) / (2^(7/180) * sqrt(5*Pi) * n^(533/900)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 08 2018

A308291 Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/(1 - x^k)^4)/k.

Original entry on oeis.org

1, 3, 6, 4, -3, -22, -23, 8, 88, 139, -19, -472, -869, -101, 2684, 5668, 2104, -15300, -37680, -22428, 86645, 252383, 202936, -482512, -1694944, -1710607, 2584008, 11368664, 13819803, -12802724, -75911328, -108463344, 53647377, 503132556, 833364427, -127320060

Offset: 1

Ilya Gutkovskiy, May 18 2019

sign

Inverse Euler transform of tetrahedral numbers (A000292).

Cf. A000292, A000335, A008683, A308290.

nmax = 36; CoefficientList[Series[Sum[MoebiusMu[k] Log[1 + x^k/(1 - x^k)^4]/k, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 50; s = ConstantArray[0, nmax]; Do[s[[j]] = j^2*(j + 1)*(j + 2)/6 - Sum[s[[d]]*(j - d)*(j - d + 1)*(j - d + 2)/6, {d, 1, j - 1}], {j, 1, nmax}]; Table[Sum[MoebiusMu[k/d]*s[[d]], {d, Divisors[k]}]/k, {k, 1, nmax}] (* Vaclav Kotesovec, Aug 10 2019 *)

-1 + Product_{n>=1} 1/(1 - x^n)^a(n) = g.f. of A000292.

A318121 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^4)).

Original entry on oeis.org

1, 1, 4, 15, 65, 269, 1205, 5325, 24064, 108849, 496790, 2275492, 10470720, 48325984, 223721404, 1038182441, 4828274432, 22497132116, 105001996350, 490816448220, 2297356108318, 10766317435860, 50511178395306, 237217429972191, 1115084064063866, 5246116796164594

Offset: 0

Ilya Gutkovskiy, Aug 18 2018

nonn

For n > 2, a(n) is the n-th term of the Euler transform of n-gonal pyramidal numbers.

N. J. A. Sloane, Transforms
Index to sequences related to pyramidal numbers

Cf. A000335, A279215, A279216, A279217, A279218, A279219, A318118.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + (n - 3) x^k)/(k (1 - x^k)^4), {k, 1, n}]], {x, 0, n}], {n, 0, 25}]

a(n) ~ c * d^n / sqrt(n), where d = 4.80064986801984997726284... and c = 0.244706939300168165858... - Vaclav Kotesovec, Aug 19 2018

A321598 a(n) = Sum_{d|n} d*binomial(d+2,3).

Original entry on oeis.org

1, 9, 31, 89, 176, 375, 589, 1049, 1516, 2384, 3147, 4823, 5916, 8437, 10406, 14105, 16474, 22380, 25271, 33264, 37810, 47683, 52901, 68183, 73301, 91100, 100174, 122197, 130356, 161750, 169137, 205593, 219162, 259242, 272714, 330524, 338144, 400719, 421686, 493424

Offset: 1

Ilya Gutkovskiy, Nov 14 2018

Inverse Möbius transform of A002417.

Amiram Eldar, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A000292, A000335, A001157, A001158, A001159, A002417, A013663, A059358, A278403.

Table[Sum[d Binomial[d + 2, 3], {d, Divisors[n]}], {n, 40}]
nmax = 40; Rest[CoefficientList[Series[Sum[x^k (1 + 3 x^k)/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[(2 DivisorSigma[2, n] + 3 DivisorSigma[3, n] + DivisorSigma[4, n])/6, {n, 40}]

a(n) = my(f = factor(n)); (sigma(f, 4) + 3*sigma(f, 3) + 2*sigma(f, 2)) / 6; \\ Amiram Eldar, Jan 02 2025

G.f.: Sum_{k>=1} x^k*(1 + 3*x^k)/(1 - x^k)^5.
G.f.: Sum_{k>=1} k*A000292(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^A000292(k)) = Sum_{n>=1} a(n)*x^n/n.
Dirichlet g.f.: (zeta(s-4) + 3*zeta(s-3) + 2*zeta(s-2))*zeta(s)/6.
a(n) = (2*sigma_2(n) + 3*sigma_3(n) + sigma_4(n))/6.
a(n) = Sum_{d|n} A002417(d).
Sum_{k=1..n} a(k) ~ zeta(5) * n^5 / 30. - Vaclav Kotesovec, Feb 02 2019

cp's OEIS Frontend

A278767 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(2*k-1)).

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A278769 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(5*k-3)/2).

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A295179 Expansion of Product_{k>=1} 1/(1 - x^k)^(3*k*(k-1)/2+1).

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A292387 Expansion of Product_{k>=1} (1 - x^k)^(k*(k+1)*(k+2)/6).

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A293551 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>=1} x^j/(j*(1 - x^j)^k)).

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A190905 Euler transform of the swinging factorial A056040.

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A302449 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(4*k^2-1)/3).

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A308291 Expansion of Sum_{k>=1} mu(k)*log(1 + x^k/(1 - x^k)^4)/k.

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A278767 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(2k-1)).

A278769 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(5k-3)/2).

A295179 Expansion of Product_{k>=1} 1/(1 - x^k)^(3k(k-1)/2+1).

A292387 Expansion of Product_{k>=1} (1 - x^k)^(k(k+1)(k+2)/6).

A302449 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(4k^2-1)/3).

A318121 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^4)).