A022604 -id:A022604 - cp's OEIS frontend

A341251 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^9.

1, 0, 9, 9, 45, 81, 201, 414, 828, 1650, 3060, 5697, 10131, 17829, 30564, 51543, 85482, 139455, 224527, 356436, 559341, 867405, 1331208, 2022525, 3044331, 4542174, 6720705, 9866794, 14377941, 20804994, 29903823, 42709860, 60631011, 85575855, 120118500, 167716548

Offset: 9

Ilya Gutkovskiy, Feb 07 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..10000

Cf. A000700, A001487, A022604, A327387, A338463, A341228, A341241, A341243, A341244, A341245, A341246, A341247.

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, 9):
seq(a(n), n=9..44);  # Alois P. Heinz, Feb 07 2021

nmax = 44; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^9, {x, 0, nmax}], x] // Drop[#, 9] &

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^9.

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

Original entry on oeis.org

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

A339735 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^9.

Original entry on oeis.org

1, -9, -9, 36, -9, 72, -9, -93, 36, 72, -9, -252, -9, 72, 72, 207, -9, -252, -9, -252, 72, 72, -9, 585, 36, 72, -93, -252, -9, -495, -9, -459, 72, 72, 72, 765, -9, 72, 72, 585, -9, -495, -9, -252, -252, 72, -9, -1278, 36, -252, 72, -252, -9, 585, 72, 585, 72, 72, -9, 1449

Offset: 1

Ilya Gutkovskiy, Dec 14 2020

sign

Cf. A022604, A316441, A339341, A339717, A339718, A339719, A339720, A339721, A339722, A339734.

a(1) = 1; a(n) = -Sum_{d|n, d < n} A339341(n/d) * a(d).

a(p^k) = A022604(k) for prime p.

cp's OEIS Frontend

A341251 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^9.

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

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A339735 Dirichlet g.f.: Product_{k>=2} 1 / (1 + k^(-s))^9.

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