A022600 -id:A022600 - cp's OEIS frontend

A341244 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^5.

1, 0, 5, 5, 15, 25, 45, 80, 125, 210, 321, 500, 745, 1110, 1620, 2326, 3315, 4660, 6500, 8955, 12261, 16640, 22425, 29990, 39870, 52701, 69230, 90460, 117620, 152225, 196066, 251455, 321195, 408710, 518060, 654317, 823690, 1033535, 1292690, 1611970, 2004462, 2485605

Offset: 5

Ilya Gutkovskiy, Feb 07 2021

nonn

Cf. A000700, A001483, A022600, A327383, A338463, A341223, A341241, A341243, A341245, A341246, A341247, A341251.

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 5):
seq(a(n), n=5..46);  # Alois P. Heinz, Feb 07 2021

nmax = 46; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^5, {x, 0, nmax}], x] // Drop[#, 5] &

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

A339720 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^5.

Original entry on oeis.org

1, -5, -5, 10, -5, 20, -5, -15, 10, 20, -5, -30, -5, 20, 20, 30, -5, -30, -5, -30, 20, 20, -5, 45, 10, 20, -15, -30, -5, -55, -5, -56, 20, 20, 20, 35, -5, 20, 20, 45, -5, -55, -5, -30, -30, 20, -5, -105, 10, -30, 20, -30, -5, 45, 20, 45, 20, 20, -5, 45, -5, 20, -30, 85, 20

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022600, A316441, A339337, A339717, A339718, A339719, A339721, A339722.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339337(n/d) * a(d).

a(p^k) = A022600(k) for prime p.

A025233 Expansion of Product_{m>=1} (1 + q^m)^48.

Original entry on oeis.org

1, 48, 1176, 19648, 252204, 2655456, 23901760, 189208704, 1344644814, 8713158928, 52107076128, 290374290624, 1519725061816, 7518508799904, 35352238216704, 158716136933504, 683059486979301, 2827559773199856

Offset: 0

N. J. A. Sloane

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A022596, A022600.

nmax = 20; CoefficientList[Series[Product[(1 + x^k)^48, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

a(n) ~ exp(4*Pi*sqrt(n)) / (2^(49/2) * n^(3/4)) * (1 + (4*Pi - 3/(32*Pi))/sqrt(n)). - Vaclav Kotesovec, Nov 10 2017

cp's OEIS Frontend

A341244 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^5.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A339720 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^5.

Views

Author

Keywords

Crossrefs

Formula

A025233 Expansion of Product_{m>=1} (1 + q^m)^48.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula