A023872 -id:A023872 - cp's OEIS frontend

A386720 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} 1/(1 - x^k)^(k^3) is the g.f. of A023872.

1, 1, 19, 163, 1571, 15276, 152029, 1525420, 15460771, 157716235, 1617959044, 16672687769, 172459185341, 1789587777849, 18621317408384, 194222638392213, 2029985619026851, 21256104343844595, 222937740908641405, 2341629730618924374, 24627719497316157396, 259326672761381979574

Offset: 0

Vaclav Kotesovec, Jul 31 2025

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..960

Cf. A023872.

Cf. A008485, A255672, A380290.

with(numtheory):
G(x) := series(exp(add(sigma[4](k)*x^k/k, k = 1..25)), x, 26):
seq(coeftayl(G(x)^n, x = 0, n), n = 0..25);

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

a(n) = [x^n] exp(n*Sum_{k >= 1} sigma_4(k)*x^k/k).

a(n) ~ c * d^n / sqrt(n), where d = 10.783710654896500462544161711323081108292517438268962307143535279238... and c = 0.2464683956609371456774144752559018514863700235623819263696832303304...

A013663 Decimal expansion of zeta(5).

Original entry on oeis.org

1, 0, 3, 6, 9, 2, 7, 7, 5, 5, 1, 4, 3, 3, 6, 9, 9, 2, 6, 3, 3, 1, 3, 6, 5, 4, 8, 6, 4, 5, 7, 0, 3, 4, 1, 6, 8, 0, 5, 7, 0, 8, 0, 9, 1, 9, 5, 0, 1, 9, 1, 2, 8, 1, 1, 9, 7, 4, 1, 9, 2, 6, 7, 7, 9, 0, 3, 8, 0, 3, 5, 8, 9, 7, 8, 6, 2, 8, 1, 4, 8, 4, 5, 6, 0, 0, 4, 3, 1, 0, 6, 5, 5, 7, 1, 3, 3, 3, 3

Offset: 1

N. J. A. Sloane

In a widely distributed May 2011 email, Wadim Zudilin gave a rebuttal to v1 of Kim's 2011 preprint: "The mistake (unfixable) is on p. 6, line after eq. (3.3). 'Without loss of generality' can be shown to work only for a finite set of n_k's; as the n_k are sufficiently large (and N is fixed), the inequality for epsilon is false." In a May 2013 email, Zudilin extended his rebuttal to cover v2, concluding that Kim's argument "implies that at least one of zeta(2), zeta(3), zeta(4) and zeta(5) is irrational, which is trivial." - Jonathan Sondow, May 06 2013

General: zeta(2*s + 1) = (A000364(s)/A331839(s)) * Pi^(2*s + 1) * Product_{k >= 1} (A002145(k)^(2*s + 1) + 1)/(A002145(k)^(2*s + 1) - 1), for s >= 1. - Dimitris Valianatos, Apr 27 2020

			1/1^5 + 1/2^5 + 1/3^5 + 1/4^5 + 1/5^5 + 1/6^5 + 1/7^5 + ... =
1 + 1/32 + 1/243 + 1/1024 + 1/3125 + 1/7776 + 1/16807 + ... = 1.036927755143369926331365486457...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 262.
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Michael J. Dancs and Tian-Xiao He, An Euler-type formula for zeta(2k+1), Journal of Number Theory, Volume 118, Issue 2, June 2006, Pages 192-199.
Robert J. Harley, Zeta(3), Zeta(5), .., Zeta(99) 10000 digits (txt, 400 KB).
Yong-Cheol Kim, zeta(5) is irrational, arXiv:1105.0730 [math.CA], 2011. [Jonathan Vos Post, May 4, 2011].
Simon Plouffe, Computation of Zeta(5)
Simon Plouffe, Zeta(5), the sum(1/n**5, n=1..infinity) to 512 digits
Simon Plouffe, Other interesting computations at numberworld.org.
Zhi-Wei Sun, A curious hypergeometric series related to Riemann's zeta function, Question 486353 at MathOverflow, Jan. 21, 2025.
Chuanan Wei, Some fast convergent series for the mathematical constants zeta(4) and zeta(5), arXiv:2303.07887 [math.CO], 2023.
Wikipedia, Zeta constant
Wadim Zudilin, One of the numbers ζ(5), ζ(7), ζ(9), ζ(11) is irrational, Russ. Math. Surv., 56 (2001), 774-776.
Index entries for constants related to zeta

Cf. A013661, A002117, A013662, A013664, A013665, A013666, A013667, A013668, A013669, A013670, A013671, A013672, A013673, A013674, A013675, A013676, A013677, A013678, A293904.
Cf. A243264, A255323.
Cf. A023872, A023873, A248882, A255050, A255052, A057528, A260404.

RealDigits[Zeta[5], 10, 100][[1]] (* Alonso del Arte, Jan 13 2012 *)

PARI
```
zeta(5) \\ Michel Marcus, Apr 17 2016
```

From Peter Bala, Dec 04 2013: (Start)
Definition: zeta(5) = Sum_{n >= 1} 1/n^5.
zeta(5) = 2^5/(2^5 - 1)*(Sum_{n even} n^5*p(n)*p(1/n)/(n^2 - 1)^6 ), where p(n) = n^2 + 3. See A013667, A013671 and A013675. (End)
zeta(5) = Sum_{n >= 1} (A010052(n)/n^(5/2)) = Sum_{n >= 1} ((floor(sqrt(n)) - floor(sqrt(n-1)))/n^(5/2)). - Mikael Aaltonen, Feb 22 2015
zeta(5) = Product_{k>=1} 1/(1 - 1/prime(k)^5). - Vaclav Kotesovec, Apr 30 2020
From Artur Jasinski, Jun 27 2020: (Start)
zeta(5) = (-1/30)*Integral_{x=0..1} log(1-x^4)^5/x^5.
zeta(5) = (1/24)*Integral_{x=0..infinity} x^4/(exp(x)-1).
zeta(5) = (2/45)*Integral_{x=0..infinity} x^4/(exp(x)+1).
zeta(5) = (1/(1488*zeta(1/2)^5))*(-5*Pi^5*zeta(1/2)^5 + 96*zeta'(1/2)^5 - 240*zeta(1/2)*zeta'(1/2)^3*zeta''(1/2) + 120*zeta(1/2)^2*zeta'(1/2)*zeta''(1/2)^2 + 80*zeta(1/2)^2*zeta'(1/2)^2*zeta'''(1/2)- 40*zeta(1/2)^3*zeta''(1/2)*zeta'''(1/2) - 20*zeta(1/2)^3*zeta'(1/2)*zeta''''(1/2)+4*zeta(1/2)^4*zeta'''''(1/2)). (End).
From Peter Bala, Oct 29 2023: (Start)
zeta(3) = (8/45)*Integral_{x >= 1} x^3*log(x)^3*(1 + log(x))*log(1 + 1/x^x) dx =  (2/45)*Integral_{x >= 1} x^4*log(x)^4*(1 + log(x))/(1 + x^x) dx.
zeta(5) = 131/128 + 26*Sum_{n >= 1} (n^2 + 2*n + 40/39)/(n*(n + 1)*(n + 2))^5.
zeta(5) =  5162893/4976640 - 1323520*Sum_{n >= 1} (n^2 + 4*n + 56288/12925)/(n*(n + 1)*(n + 2)*(n + 3)*(n + 4))^5. Taking 10 terms of the series gives a value for zeta(5) correct to 20 decimal places.
Conjecture: for k >= 1, there exist rational numbers A(k), B(k) and c(k) such that zeta(5) = A(k) + B(k)*Sum_{n >= 1} (n^2 + 2*k*n + c(k))/(n*(n + 1)*...*(n + 2*k))^5. A similar conjecture can be made for the constant zeta(3). (End)
zeta(5) = (694/204813)*Pi^5 - Sum_{n >= 1} (6280/3251)*(1/(n^5*(exp(4*Pi*n)-1))) + Sum_{n >= 1} (296/3251)*(1/(n^5*(exp(5*Pi*n)-1))) - Sum_{n >= 1} (1073/6502)*(1/(n^5*(exp(10*Pi*n)-1))) + Sum_{n >= 1} (37/6502)*(1/(n^5*(exp(20*Pi*n)-1))). - Simon Plouffe, Jan 06 2024
From Peter Bala, Apr 27 2025: (Start)
zeta(5) = 1/5! * Integral_{x >= 0} x^5 * exp(x)/(exp(x) - 1)^2 dx = (16/15) * 1/5! * Integral_{x >= 0} x^5 * exp(x)/(exp(x) + 1)^2 dx.
zeta(5) = 1/6! * Integral_{x >= 0} x^6 * exp(x)*(exp(x) + 1)/(exp(x) - 1)^3 dx = 1/(3^3 * 5^2) * Integral_{x >= 0} x^6 * exp(x)*(exp(x) - 1)/(exp(x) + 1)^3 dx. (End)
zeta(5) = Sum_{i, j >= 1} 1/((i^4)*j*binomial(i+j, i)). More generally, zeta(n+1) = Sum_{i, j >= 1} 1/((i^n)*j*binomial(i+j, i)) for n >= 1. - Peter Bala, Aug 07 2025

A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

Original entry on oeis.org

1, 1, 5, 14, 40, 101, 266, 649, 1593, 3765, 8813, 20168, 45649, 101591, 223654, 486046, 1045541, 2225167, 4692421, 9804734, 20318249, 41766843, 85218989, 172628766, 347338117, 694330731, 1379437080, 2724353422, 5350185097, 10449901555, 20304465729, 39254599832

Offset: 0

Olivier Gérard

In general, if g.f. = Product_{k>=1} 1/(1 - x^k)^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(4*Pi * c2^(1/4) * n^(3/4) / (3*15^(1/4)) + c1*Zeta(3) / Pi^2 * sqrt(15*n/c2) + (Pi * 5^(1/4) * c0 / (2*3^(3/4) * c2^(1/4)) - 15^(5/4) * c1^2 * Zeta(3)^2 / (2*c2^(5/4) * Pi^5)) * n^(1/4) + c1/12 + 75 * c1^3 * Zeta(3)^3 / (c2^2 * Pi^8) - 5*c0 * c1 * Zeta(3) / (4*c2 * Pi^2) - c2*Zeta(3) / (4*Pi^2)) * Pi^(c1/12) * (c2/15)^(1/8 + c0/8 + c1/48) / (A^c1 * 2^((c0 + 3)/2) * n^(5/8 + c0/8 + c1/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017
Let A(x) = Product_{k >= 1} (1 - x^k)^(-k^2). The sequence defined by u(n) := [x^n] A(x)^n is conjectured to satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 7 and all positive integers n and r. See A380290. - Peter Bala, Feb 02 2025
a(n) is the number of partitions of n where there are k^2 sorts of part k. - Joerg Arndt, Feb 02 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from Alois P. Heinz)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
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. 21.

Euler transform of squares (A000290).
Cf. A000219, A023872-A023878, A294530, A380290.
Column k=2 of A144048. - Alois P. Heinz, Nov 02 2012

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^2: k in [1..m]]) )); // G. C. Greubel, Oct 29 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^2, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35); # Alois P. Heinz, Nov 02 2012

max = 31; Series[ Product[ 1/(1-x^k)^k^2, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Mar 05 2013 *)

m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^2)) \\ G. C. Greubel, Oct 29 2018

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^2)
print([b(n) for n in range(32)]) # Peter Luschny, Nov 11 2020

a(n) = 1/n * Sum_{k=1..n} a(n-k)*sigma_3(k), n > 0, a(0)=1, where sigma_3(n) = A001158(n) = sum of cubes of divisors of n. - Vladeta Jovovic, Jan 20 2002
G.f.: Prod_{n>=1} exp(sigma_3(n)*x^n/n), where sigma_3(n) is the sum of cubes of divisors of n (=A001158(n)). - N-E. Fahssi, Mar 28 2010
G.f. (conjectured): 1/Product_{n>=1} E(x^n)^J2(n) where E(x) = Product_{n>=1} 1-x^n and J2(n) = A007434(n) [follows from the identity Sum_{d|n} J2(d) = n^2 - Peter Bala, Feb 02 2025]. - Joerg Arndt, Jan 25 2011
a(n) ~ exp(4 * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) / (4*Pi^2)) / (2^(3/2) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 27 2015

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A259068 Decimal expansion of zeta'(-3) (the derivative of Riemann's zeta function at -3).

Original entry on oeis.org

0, 0, 5, 3, 7, 8, 5, 7, 6, 3, 5, 7, 7, 7, 4, 3, 0, 1, 1, 4, 4, 4, 1, 6, 9, 7, 4, 2, 1, 0, 4, 1, 3, 8, 4, 2, 8, 9, 5, 6, 6, 4, 4, 3, 9, 7, 4, 2, 2, 9, 5, 5, 0, 7, 0, 5, 9, 4, 4, 7, 0, 2, 3, 2, 2, 3, 3, 2, 4, 5, 0, 1, 9, 9, 7, 9, 2, 4, 0, 6, 9, 5, 8, 6, 0, 9, 5, 1, 0, 3, 8, 7, 0, 8, 2, 5, 6, 8, 3, 2, 6, 7, 1, 2, 2, 4, 3

Offset: 0

Jean-François Alcover, Jun 18 2015

			0.0053785763577743011444169742104138428956644397422955070594470232233245...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15.1 Generalized Glaisher constants, p. 136-137.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's MathWorld, Riemann Zeta Function.
Wikipedia, Riemann Zeta Function
Index entries for constants related to zeta

Cf. A000335, A000391, A000417, A000428, A023872, A057527, A057528, A255050, A255052, A258350, A258351, A258352, A260404.

Join[{0, 0}, RealDigits[Zeta'[-3], 10, 105] // First]

zeta'(-n) = (BernoulliB(n+1)*HarmonicNumber(n))/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant, that is the n-th generalized Glaisher constant.
zeta'(-3) = -11/720 - log(A(3)), where A(3) is A243263.
Equals -11/720 + (gamma + log(2*Pi))/120 - 3*Zeta'(4)/(4*Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 24 2015

A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

Original entry on oeis.org

1, 1, 8, 35, 119, 433, 1476, 4962, 16128, 51367, 160105, 490219, 1476420, 4378430, 12805008, 36962779, 105417214, 297265597, 829429279, 2291305897, 6270497702, 17008094490, 45744921052, 122052000601, 323166712109, 849453194355, 2217289285055, 5749149331789

Offset: 0

Vaclav Kotesovec, Mar 05 2015

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..5510 (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. 22.

Cf. A023872, A026007, A027998, A248883, A248884, A309335.

Column k=3 of A284992.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[(1+x^k)^k^3: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n) option remember; add(
      (-1)^(n/d+1)*d^4, d=numtheory[divisors](n))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(b(k)*a(n-k), k=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 16 2017

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

x = 'x + O('x^50); Vec(prod(k=1, 50, (1 + x^k)^(k^3))) \\ Indranil Ghosh, Apr 06 2017

a(n) ~ Zeta(5)^(1/10) * 3^(1/5) * exp(2^(-11/5) * 3^(2/5) * 5^(6/5) * Zeta(5)^(1/5) * n^(4/5)) / (2^(71/120) * 5^(2/5)* sqrt(Pi) * n^(3/5)), where Zeta(5) = A013663.
a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A284900(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k*(1 + 4*x^k + x^(2*k))/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 30 2018
Euler transform of A309335. - Georg Fischer, Nov 10 2020

A144048 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j^k).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 9, 14, 13, 7, 1, 1, 17, 36, 40, 24, 11, 1, 1, 33, 98, 136, 101, 48, 15, 1, 1, 65, 276, 490, 477, 266, 86, 22, 1, 1, 129, 794, 1828, 2411, 1703, 649, 160, 30, 1, 1, 257, 2316, 6970, 12729, 11940, 5746, 1593, 282, 42, 1, 1, 513

Offset: 0

Alois P. Heinz, Sep 08 2008

In general, column k > 0 is asymptotic to (Gamma(k+2)*Zeta(k+2))^((1-2*Zeta(-k)) /(2*k+4)) * exp((k+2)/(k+1) * (Gamma(k+2)*Zeta(k+2))^(1/(k+2)) * n^((k+1)/(k+2)) + Zeta'(-k)) / (sqrt(2*Pi*(k+2)) * n^((k+3-2*Zeta(-k))/(2*k+4))). - Vaclav Kotesovec, Mar 01 2015

			Square array begins:
  1,  1,   1,   1,    1,     1, ...
  1,  1,   1,   1,    1,     1, ...
  2,  3,   5,   9,   17,    33, ...
  3,  6,  14,  36,   98,   276, ...
  5, 13,  40, 136,  490,  1828, ...
  7, 24, 101, 477, 2411, 12729, ...

Alois P. Heinz, Antidiagonals = 0..99, flattened
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. 21.
N. J. A. Sloane, Transforms

Columns k=0-9 give: A000041, A000219, A023871, A023872, A023873, A023874, A023875, A023876, A023877, A023878.
Rows give: 0-1: A000012, 2: A000051, A094373, 3: A001550, 4: A283456, 5: A283457.
Main diagonal gives A252782.
Cf. A283272.

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->j^k)(n); seq(seq(A(n,d-n), n=0..d), d=0..13);

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[Function[j, j^k]][n]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

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

A206623 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^3).

Original entry on oeis.org

1, 2, 18, 88, 398, 1768, 7508, 30644, 121310, 467234, 1756080, 6457168, 23274788, 82381584, 286760344, 982874120, 3320800590, 11070619228, 36446345198, 118581503192, 381552358872, 1214868568728, 3829841265428, 11959828895612, 37013411304892, 113570015855642

Offset: 0

Paul D. Hanna, Feb 12 2012

nonn

Convolution of A023872 and A248882. - Vaclav Kotesovec, Aug 19 2015

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 88*x^3 + 398*x^4 + 1768*x^5 + 7508*x^6 +...
where A(x) = (1+x)/(1-x) * (1+x^2)^8/(1-x^2)^8 * (1+x^3)^27/(1-x^3)^27 *...
Also, A(x) = Euler transform of [2,15,54,120,250,405,686,960,1458,...]:
A(x) = 1/((1-x)^2*(1-x^2)^15*(1-x^3)^54*(1-x^4)^120*(1-x^5)^250*(1-x^6)^405*...).

Seiichi Manyama, Table of n, a(n) for n = 0..4595 (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. 23.

Cf. A156616, A206622, A206624, A001159 (sigma_4).

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)

{a(n)=polcoeff(prod(m=1,n+1,((1+x^m)/(1-x^m+x*O(x^n)))^(m^3)),n)}

{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 4)-sigma(m, 4))/8*x^m/m)+x*O(x^n)), n)}

{a(n)=local(InvEulerGF=x*(2+15*x+46*x^2+60*x^3+46*x^4+15*x^5+2*x^6)/(1-x^2+x*O(x^n))^4);polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^polcoeff(InvEulerGF,k)),n)}
for(n=0,30,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} (sigma_4(2*n) - sigma_4(n))/8 * x^n/n ), where sigma_4(n) is the sum of 4th powers of divisors of n (A001159).
Inverse Euler transform has g.f.: x*(2 + 15*x + 46*x^2 + 60*x^3 + 46*x^4 + 15*x^5 + 2*x^6)/(1-x^2)^4.
a(n) ~ (93*Zeta(5))^(59/600) * exp(5/4 * (93*Zeta(5)/2)^(1/5) * n^(4/5) + Zeta'(-3)) / (2^(59/100) * sqrt(5*Pi) * n^(359/600)), where Zeta(5) = A013663, Zeta'(-3) = A259068. - Vaclav Kotesovec, Aug 19 2015

A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)/6).

Original entry on oeis.org

1, 0, 0, 1, 4, 10, 21, 39, 76, 145, 294, 581, 1169, 2276, 4435, 8494, 16237, 30768, 58221, 109466, 205223, 382658, 710808, 1314091, 2420437, 4439753, 8115645, 14781062, 26833241, 48550863, 87575527, 157480827, 282362462, 504819198, 900058558, 1600424247

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

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

Cf. A000335, A023872, A258346, A258347, A258348, A258349, A258350, A258351.

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

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n, 3))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) ~ Zeta(5)^(379/3600) / (2^(521/1800) * sqrt(5*Pi) * n^(2179/3600)) * exp(Zeta'(-1)/3 + Zeta(3)/(8*Pi^2) - Pi^16 / (3110400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (216000 * Zeta(5)^2) - Zeta(3)^2/(90*Zeta(5)) + Zeta'(-3)/6 + (-Pi^12 / (10800000 * 2^(2/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (900 * 2^(2/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (36000 * 2^(4/5) * Zeta(5)^(7/5)) + Zeta(3) / (3 * 2^(4/5) * Zeta(5)^(2/5))) * n^(2/5) - Pi^4 / (180 * 2^(1/5) * Zeta(5)^(3/5)) * n^(3/5) + 5 * Zeta(5)^(1/5) / 2^(8/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

A283264 Expansion of exp( Sum_{n>=1} -sigma_4(n)*x^n/n ) in powers of x.

Original entry on oeis.org

1, -1, -8, -19, -9, 127, 500, 1038, 448, -4967, -21463, -50043, -59084, 70418, 600080, 1837349, 3532062, 3179251, -6965009, -42260393, -119597290, -224546234, -223670132, 292245783, 2156083245, 6428174973, 13030612271, 16820582355, -133402359, -78307103593

Offset: 0

Seiichi Manyama, Mar 04 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Column k=3 of A283272.
Cf. A023872 (exp( Sum_{n>=1} sigma_4(n)*x^n/n )).
Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), this sequence (k=4), A283271 (k=5).

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: -n^3)
print([b(n) for n in range(30)]) # Peter Luschny, Nov 11 2020

G.f.: Product_{n>=1} (1 - x^n)^(n^3).

a(n) = -(1/n)*Sum_{k=1..n} sigma_4(k)*a(n-k).

A258350 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k+1)(k+2)).

Original entry on oeis.org

1, 6, 45, 260, 1410, 7026, 33212, 149190, 643959, 2681020, 10820736, 42468828, 162566956, 608302638, 2229485529, 8016901068, 28324233846, 98447346282, 336996263702, 1137220855428, 3786525025002, 12449461237388, 40446207528429, 129926295916884, 412912082761651

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

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

Cf. A023872, A000335, A258342, A258347, A258348, A258349, A258351, A258352.

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

a(n) ~ (3*Zeta(5))^(79/600) / (2^(21/200) * sqrt(5*Pi) * n^(379/600)) * exp(2*Zeta'(-1) - 3*Zeta(3)/(4*Pi^2) - Pi^16 / (518400000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3) + (Pi^12 / (1800000 * 2^(3/5) * 3^(1/5) * Zeta(5)^(11/5)) - Pi^4 * Zeta(3) / (150 * 2^(3/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(1/5) * 3^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(1/5) * (3*Zeta(5))^(2/5))) * n^(2/5) + Pi^4 / (30 * 2^(4/5) * (3*Zeta(5))^(3/5)) * n^(3/5) + 5 * (3*Zeta(5))^(1/5) / 2^(7/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663, Zeta'(-1) = A084448 = 1/12 - log(A074962), Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4.

cp's OEIS Frontend

A386720 a(n) = [x^n] G(x)^n, where G(x) = Product_{k >= 1} 1/(1 - x^k)^(k^3) is the g.f. of A023872.

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A013663 Decimal expansion of zeta(5).

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A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

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A259068 Decimal expansion of zeta'(-3) (the derivative of Riemann's zeta function at -3).

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A248882 Expansion of Product_{k>=1} (1+x^k)^(k^3).

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A144048 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is Euler transform of (j->j^k).

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A206623 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^3).

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A258352 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k-1)(k-2)/6).

A258350 Expansion of Product_{k>=1} 1/(1-x^k)^(k(k+1)(k+2)).