A022605 -id:A022605 - cp's OEIS frontend

A022600 Expansion of Product_{m>=1} (1+q^m)^(-5).

1, -5, 10, -15, 30, -56, 85, -130, 205, -315, 465, -665, 960, -1380, 1925, -2651, 3660, -5020, 6775, -9070, 12126, -16115, 21220, -27765, 36235, -47101, 60810, -78115, 100105, -127825, 162391, -205530, 259475, -326565

Offset: 0

N. J. A. Sloane

sign

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

Cf. Related to Expansion of Product_{m>=1} (1+q^m)^k: A022627 (k=-32), A022626 (k=-31), A022625 (k=-30), A022624 (k=-29), A022623 (k=-28), A022622 (k=-27), A022621 (k=-26), A022620 (k=-25), A007191 (k=-24), A022618 (k=-23), A022617 (k=-22), A022616 (k=-21), A022615 (k=-20), A022614 (k=-19), A022613 (k=-18), A022612 (k=-17), A022611 (k=-16), A022610 (k=-15), A022609 (k=-14), A022608 (k=-13), A007249 (k=-12), A022606 (k=-11), A022605 (k=-10), A022604 (k=-9), A007259 (k=-8), A022602 (k=-7), A022601 (k=-6), this sequence (k=-5), A022599 (k=-4), A022598 (k=-3), A022597 (k=-2), A081362 (k=-1), A000009 (k=1), A022567 (k=2), A022568 (k=3), A022569 (k=4), A022570 (k=5), A022571 (k=6), A022572 (k=7), A022573 (k=8), A022574 (k=9), A022575 (k=10), A022576 (k=11), A022577 (k=12), A022578 (k=13), A022579 (k=14), A022580 (k=15), A022581 (k=16), A022582 (k=17), A022583 (k=18), A022584 (k=19), A022585 (k=20), A022586 (k=21), A022587 (k=22), A022588 (k=23), A014103 (k=24), A022589 (k=25), A022590 (k=26), A022591 (k=27), A022592 (k=28), A022593 (k=29), A022594 (k=30), A022595 (k=31), A022596 (k=32), A025233 (k=48).

Column k=5 of A286352.

nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)

x='x+O('x^50); Vec(prod(m=1, 50, (1 + x^m)^(-5))) \\ Indranil Ghosh, Apr 05 2017

a(n) ~ (-1)^n * 5^(1/4) * exp(Pi*sqrt(5*n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-5*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A341253 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^10.

Original entry on oeis.org

1, 0, 10, 10, 55, 100, 265, 560, 1175, 2420, 4667, 9000, 16575, 30180, 53470, 93152, 159395, 268190, 444910, 727360, 1174563, 1873320, 2955010, 4611960, 7127305, 10912244, 16560430, 24924550, 37217620, 55160650, 81174270, 118651560, 172316445, 248718830, 356892660

Offset: 10

Ilya Gutkovskiy, Feb 07 2021

nonn

Cf. A000700, A001488, A022605, A327388, A338463, A341236, A341241, A341243, A341244, 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, 10):
seq(a(n), n=10..44);  # Alois P. Heinz, Feb 07 2021

nmax = 44; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^10, {x, 0, nmax}], x] // Drop[#, 10] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^10.

cp's OEIS Frontend

A022600 Expansion of Product_{m>=1} (1+q^m)^(-5).

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