A278768 -id:A278768 - cp's OEIS frontend

A294591 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(3k-1)/2)(1 - x^(2k))^(k(3*k+1)/2)).

1, 1, 3, 8, 18, 40, 88, 184, 384, 783, 1573, 3110, 6087, 11745, 22450, 42466, 79597, 147890, 272632, 498696, 905846, 1634270, 2929804, 5220581, 9249440, 16297659, 28567571, 49825296, 86487331, 149438681, 257077485, 440378787, 751313413, 1276765557, 2161511352

Offset: 0

Ilya Gutkovskiy, Nov 05 2017

nonn

Euler transform of the generalized pentagonal numbers (A001318).

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000294, A001318, A095699, A278768, A294654, A294655, A294667, A294669.

nmax = 34; CoefficientList[Series[Product[1/((1 - x^(2 k - 1))^(k (3 k - 1)/2) (1 - x^(2 k))^(k (3 k + 1)/2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d Ceiling[d/2] Ceiling[(3 d + 1)/2]/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 34}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A001318(k).

a(n) ~ exp(Pi * 2^(5/4) / (3*5^(1/4)) * n^(3/4) + 3*Zeta(3) * sqrt(5*n) / (2^(3/2) * Pi^2) + (Pi/48 - 45*Zeta(3)^2 / (8*Pi^5)) * (5*n/2)^(1/4) + 225*Zeta(3)^3 / (8*Pi^8) - 11*Zeta(3) / (64*Pi^2))/ (2^(95/48) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 07 2017

A294667 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k+1)/2).

Original entry on oeis.org

1, 2, 10, 33, 110, 332, 997, 2829, 7889, 21299, 56400, 146028, 371681, 929498, 2290296, 5562369, 13336036, 31583177, 73957845, 171342592, 393018517, 893000610, 2011039286, 4490680381, 9947577333, 21867539862, 47721817473, 103420870299, 222641160569

Offset: 0

Vaclav Kotesovec, Nov 06 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A294591, A278768.

nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^(k*(3*k+1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi * 2^(7/4) * n^(3/4) / (3*5^(1/4)) + Zeta(3) * sqrt(5*n) / (sqrt(2) * Pi^2) - 5^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(7/4) * Pi^5) + (25 * Zeta(3)^3) / (6*Pi^8) - 3*Zeta(3) / (8*Pi^2) + 1/24) * Pi^(1/24) / (sqrt(A) * 2^(157/96) * 5^(13/96) * n^(61/96)), where A is the Glaisher-Kinkelin constant A074962.

A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-2)).

Original entry on oeis.org

1, 1, 9, 30, 106, 339, 1106, 3355, 10102, 29358, 83908, 234394, 644286, 1739933, 4631675, 12153197, 31485413, 80576160, 203902261, 510490213, 1265353568, 3106771717, 7559844833, 18239351931, 43650061720, 103657177941, 244346681972, 571930478187, 1329655624297, 3071230379625, 7049750442386, 16085170634548, 36489192684910

Offset: 0

Ilya Gutkovskiy, Nov 30 2016

Euler transform of the octagonal numbers (A000567).

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
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, Octagonal Number
Index to sequences related to polygonal numbers

Cf. A000294, A000567, A000335, A023871, A278768, A294667, A294691, A294692.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(3*d-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 (3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]

from sympy import divisors
from sympy.core.cache import cacheit
@cacheit
def a(n): return 1 if n==0 else sum(sum(d**2*(3*d - 2) for d in divisors(j))*a(n - j) for j in range(1, n + 1))//n
print([a(n) for n in range(51)]) # Indranil Ghosh, Aug 06 2017, after Maple code

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

a(n) ~ exp(4*Pi*n^(3/4) / (3*5^(1/4)) - 2*Zeta(3) * sqrt(5*n) / Pi^2 - 10*Zeta(3)^2 * (5*n)^(1/4) / Pi^5 - 200*Zeta(3)^3 / (3*Pi^8) - 3*Zeta(3) / (4*Pi^2) - 1/6) * A^2 / (2^(3/2) * 5^(1/12) * Pi^(1/6) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^3)).

Original entry on oeis.org

1, 1, 3, 10, 40, 150, 616, 2456, 10102, 41400, 171526, 712111, 2972115, 12434993, 52195414, 219567909, 925704792, 3909841659, 16541598215, 70085877919, 297347922785, 1263046810334, 5370930049915, 22861883482838, 97402827429118, 415332438952517, 1772380322197432

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 numbers.

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
Index to sequences related to polygonal numbers

Cf. A000294, A023871, A274998, A278767, A278768, A278769.

Table[SeriesCoefficient[Exp[Sum[x^k (1 + (n - 3) x^k)/(k (1 - x^k)^3), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

a(n) ~ c * d^n / sqrt(n), where d = 4.3505530790182509701639869563721679988879373943131559534408716195123... and c = 0.2276354216252041005336767937139336687746108521151301186102034... - Vaclav Kotesovec, Aug 18 2018

A294669 Expansion of Product_{k>=1} 1/(1 - x^(2k-1))^(k(3*k-1)/2).

Original entry on oeis.org

1, 1, 1, 6, 6, 18, 33, 55, 115, 185, 373, 604, 1113, 1903, 3251, 5678, 9350, 16153, 26420, 44561, 72912, 120150, 196329, 317988, 516881, 827778, 1333570, 2120492, 3381947, 5347513, 8447482, 13285450, 20813814, 32547272, 50638328, 78707858, 121738479

Offset: 0

Vaclav Kotesovec, Nov 06 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A035528, A262811, A294591, A278768.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(k*(3*k-1)/2),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ exp(2*Pi * n^(3/4) / (3*5^(1/4)) + Zeta(3) * sqrt(5*n) / Pi^2 + 5^(1/4) * (Pi/48 - 5*Zeta(3)^2 / Pi^5) * n^(1/4) + 100*Zeta(3)^3 / (3*Pi^8) + 17*Zeta(3) / (96*Pi^2) - 1/24) * sqrt(A) / (2^(101/48) * 5^(11/96) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

A294692 Expansion of Product_{k>=1} 1 / (1 - x^k)^(k(3k + 2)).

Original entry on oeis.org

1, 5, 31, 148, 667, 2754, 10823, 40393, 145085, 502780, 1690603, 5530649, 17658430, 55141520, 168751779, 506933980, 1496999360, 4350994324, 12460305177, 35192973824, 98116587875, 270220568883, 735668636567, 1981082952258, 5279879097853, 13933764841202

Offset: 0

Vaclav Kotesovec, Nov 07 2017

nonn

Robert Israel, Table of n, a(n) for n = 0..5000

Cf. A074962, A274998, A278768, A294655, A294667.

N:= 50:
S:= series(mul(1/(1-x^k)^(k*(3*k+2)), k=1..N),x,N+1):
seq(coeff(S,x,n),n=0..N); # Robert Israel, Nov 07 2017

nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^(k*(3*k+2)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(4*Pi*n^(3/4) / (3*5^(1/4)) + 2*Zeta(3) * sqrt(5*n) / Pi^2 - 2*5^(5/4) * Zeta(3)^2 * n^(1/4) / Pi^5 + 200*Zeta(3)^3 / (3*Pi^8) - 3*Zeta(3) / (4*Pi^2) + 1/6) * Pi^(1/6) / (A^2 * 2^(3/2) * 5^(1/6) * n^(2/3)), where A is the Glaisher-Kinkelin constant A074962.

A317017 Expansion of Product_{k>=1} 1/(1 - x^k)^((3k+1)binomial(k+2,3)/4).

Original entry on oeis.org

1, 1, 8, 33, 126, 441, 1571, 5338, 17900, 58359, 187134, 588966, 1826537, 5580784, 16831549, 50135506, 147650112, 430187724, 1240908651, 3545808444, 10042128414, 28201458999, 78567720054, 217225969695, 596254164090, 1625343030654, 4401332943214, 11843216471115, 31674767502610

Offset: 0

Ilya Gutkovskiy, Jul 19 2018

nonn

Euler transform of A001296.

N. J. A. Sloane, Transforms

Cf. A000391, A001296, A278768, A279216, A305653, A317019, A317020, A317021.

a:=series(mul(1/(1-x^k)^((3*k+1)*binomial(k+2,3)/4),k=1..100),x=0,29): seq(coeff(a,x,n),n=0..28); # Paolo P. Lava, Apr 02 2019

nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^((3 k + 1) Binomial[k + 2, 3]/4), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[Exp[Sum[x^k (1 + 2 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (3 d + 1)/24, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]

G.f.: Product_{k>=1} 1/(1 - x^k)^A001296(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^k)^5)).
a(n) ~ Pi^(1/288)/(2 * 3^(577/864) * 7^(145/1728) * n^(1009/1728)) * exp(1/144 - (1/12-Zeta'(-1))/12 - (11 * Zeta(3))/(80 * Pi^2) + (1383 * Zeta(5))/(640 * Pi^4) + (11025 * Zeta(3) * Zeta(5)^2)/(2 * Pi^12) - (694575 * Zeta(5)^3)/(2 * Pi^14) + (13127467500 * Zeta(5)^5)/Pi^24 + (5 * Zeta'(-3))/12 + ((-21 * 3^(1/3) * 7^(1/6) * Pi)/6400 - (35 * 3^(1/3) * 7^(1/6) * Zeta(3) * Zeta(5))/(2 * Pi^7) + (15435 * 3^(1/3) * 7^(1/6) * Zeta(5)^2)/(16 * Pi^9) - (175573125 * 3^(1/3) * 7^(1/6) * Zeta(5)^4)/(4 * Pi^19)) * n^(1/6) + (((7/3)^(1/3) * Zeta(3))/(4 * Pi^2) - (21 * 3^(2/3) * 7^(1/3) * Zeta(5))/(8 * Pi^4) + (147000 * 3^(2/3) * 7^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7) * Pi)/40 - (1575 * sqrt(7) * Zeta(5)^2)/Pi^9) * sqrt(n) + ((15 * 3^(1/3) * 7^(2/3) * Zeta(5))/(2 * Pi^4)) * n^(2/3) + ((2 * 3^(2/3) * Pi)/(5 * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018

cp's OEIS Frontend

A294591 Expansion of Product_{k>=1} 1/((1 - x^(2*k-1))^(k*(3*k-1)/2)*(1 - x^(2*k))^(k*(3*k+1)/2)).

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A294667 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k+1)/2).

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A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k-2)).

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A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k*(1 + (n - 3)*x^k)/(k*(1 - x^k)^3)).

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A294669 Expansion of Product_{k>=1} 1/(1 - x^(2*k-1))^(k*(3*k-1)/2).

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A294692 Expansion of Product_{k>=1} 1 / (1 - x^k)^(k*(3*k + 2)).

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A317017 Expansion of Product_{k>=1} 1/(1 - x^k)^((3*k+1)*binomial(k+2,3)/4).

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A294591 Expansion of Product_{k>=1} 1/((1 - x^(2k-1))^(k(3k-1)/2)(1 - x^(2k))^(k(3*k+1)/2)).

A294667 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k+1)/2).

A274998 Expansion of Product_{k>=1} 1/(1 - x^k)^(k(3k-2)).

A318118 a(n) = [x^n] exp(Sum_{k>=1} x^k(1 + (n - 3)x^k)/(k*(1 - x^k)^3)).

A294669 Expansion of Product_{k>=1} 1/(1 - x^(2k-1))^(k(3*k-1)/2).

A294692 Expansion of Product_{k>=1} 1 / (1 - x^k)^(k(3k + 2)).

A317017 Expansion of Product_{k>=1} 1/(1 - x^k)^((3k+1)binomial(k+2,3)/4).