A309335 -id:A309335 - cp's OEIS frontend

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

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

A309336 a(n) = n^4 if n odd, 15*n^4/16 if n even.

Original entry on oeis.org

0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561, 9375, 14641, 19440, 28561, 36015, 50625, 61440, 83521, 98415, 130321, 150000, 194481, 219615, 279841, 311040, 390625, 428415, 531441, 576240, 707281, 759375, 923521, 983040, 1185921, 1252815, 1500625, 1574640, 1874161, 1954815

Offset: 0

Ilya Gutkovskiy, Jul 24 2019

Moebius transform of A285989.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,5,0,-10,0,10,0,-5,0,1).

Cf. A000583, A016756, A026741, A059377, A285989, A308422, A309335.

a[n_] := If[OddQ[n], n^4, 15 n^4/16]; Table[a[n], {n, 0, 38}]
nmax = 38; CoefficientList[Series[x (1 + 15 x + 76 x^2 + 165 x^3 + 230 x^4 + 165 x^5 + 76 x^6 + 15 x^7 + x^8)/(1 - x^2)^5, {x, 0, nmax}], x]
LinearRecurrence[{0, 5, 0, -10, 0, 10, 0, -5, 0, 1}, {0, 1, 15, 81, 240, 625, 1215, 2401, 3840, 6561}, 39]
Table[n^4 (31 - (-1)^n)/32, {n, 0, 38}]

G.f.: x * (1 + 15*x + 76*x^2 + 165*x^3 + 230*x^4 + 165*x^5 + 76*x^6 + 15*x^7 + x^8)/(1 - x^2)^5.
G.f.: Sum_{k>=1} J_4(k) * x^k/(1 - x^(2*k)), where J_4() is the Jordan function (A059377).
Dirichlet g.f.: zeta(s-4) * (1 - 1/2^s).
a(n) = n^4 * (31 - (-1)^n)/32.
a(n) = Sum_{d|n, n/d odd} J_4(d).
Sum_{n>=1} 1/a(n) = 241*Pi^4/21600 = 1.086832913851601267313987...
Multiplicative with a(2^e) = 15*2^(4*e-4), and a(p^e) = p^(4*e) for odd primes p. - Amiram Eldar, Oct 26 2020

cp's OEIS Frontend

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

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula