A363578 -id:A363578 - cp's OEIS frontend

A363580 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 + 2*x^k)) ).

1, 1, 0, 2, 0, 2, 1, 6, -2, 11, -1, 30, -21, 76, -60, 223, -245, 653, -817, 2031, -2935, 6521, -10067, 21455, -35425, 72152, -123756, 246752, -436854, 855852, -1546777, 3001811, -5513604, 10630676, -19747742, 37949424, -71115077, 136415279, -257301742, 493313335

Offset: 0

Seiichi Manyama, Jun 10 2023

sign

Cf. A052855, A198518, A363545, A363546, A363581.

Cf. A363578.

seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, subst(A, x, x^k)*x^k/(k*(1+2*x^k)))+x*O(x^n))); Vec(A);

A(x) = (1 + 2*x) * B(x) where B(x) is the g.f. of A363578.

a(n) = A363578(n) + 2*A363578(n-1) for n > 0.

A363579 G.f. satisfies A(x) = exp( Sum_{k>=1} ((-3)^k + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, -2, 5, -11, 27, -70, 188, -502, 1355, -3712, 10269, -28546, 79777, -224153, 632581, -1791644, 5091109, -14510079, 41464784, -118773034, 340950420, -980660721, 2825700987, -8155455450, 23573749136, -68236663474, 197774787066, -573915774310, 1667300177595

Offset: 0

Seiichi Manyama, Jun 10 2023

sign

Cf. A001678, A036249, A362389, A363541, A363578.

Cf. A363581.

seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, ((-3)^k+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);

A(x) = B(x)/(1 + 3*x) where B(x) is the g.f. of A363581.
A(x) = Sum_{k>=0} a(k) * x^k = 1/(1+3*x) * 1/Product_{k>=0} (1-x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( (-3)^k + Sum_{d|k} d * a(d-1) ) * a(n-k).

cp's OEIS Frontend

A363580 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 + 2*x^k)) ).

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