A327739 -id:A327739 - cp's OEIS frontend

A327798 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*(j + 1))).

1, 0, 1, 1, 3, 3, 9, 10, 25, 34, 72, 106, 215, 330, 635, 1025, 1899, 3141, 5713, 9602, 17213, 29292, 51982, 89149, 157249, 271027, 476037, 823386, 1442063, 2500015, 4370386, 7588146, 13248591, 23026728, 40169991, 69865026, 121811765, 211954826, 369412910

Offset: 0

Ilya Gutkovskiy, Sep 25 2019

nonn

Invert transform of A032741.

Robert Israel, Table of n, a(n) for n = 0..3000

Cf. A032741, A129921, A318783, A327739.

N:= 100: # for a(0)..a(N)
G:= 1/(1-add(x^(2*k)/(1-x^k),k=1..(N+1)/2)):
S:= series(G,x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Jan 10 2023

nmax = 38; CoefficientList[Series[1/(1 - Sum[x^(2 k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(DivisorSigma[0, k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 38}]

G.f.: 1 / (1 - Sum_{k>=1} x^(2*k) / (1 - x^k)).

a(0) = 1; a(n) = Sum_{k=1..n} A032741(k) * a(n-k).

A327799 Expansion of 1 / (1 + Sum_{i>=1} Sum_{j=1..i} x^(i*j)).

Original entry on oeis.org

1, -1, 0, 0, -1, 2, -2, 2, -1, -2, 5, -6, 5, -1, -5, 10, -14, 14, -5, -10, 26, -38, 36, -15, -20, 60, -91, 93, -51, -33, 138, -223, 237, -145, -52, 307, -528, 596, -412, -43, 674, -1258, 1492, -1126, 84, 1442, -2938, 3687, -3034, 680, 3000, -6818, 9050, -7997

Offset: 0

Ilya Gutkovskiy, Sep 25 2019

sign

Cf. A038548, A159933, A327739.

nmax = 53; CoefficientList[Series[1/(1 + Sum[x^(k^2)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[Floor[(DivisorSigma[0, k] + 1)/2] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 53}]

G.f.: 1 / (1 + Sum_{k>=1} x^(k^2) / (1 - x^k)).

a(0) = 1; a(n) = -Sum_{k=1..n} A038548(k) * a(n-k).

cp's OEIS Frontend

A327798 Expansion of 1 / (1 - Sum_{i>=1, j>=1} x^(i*(j + 1))).

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