A363567 -id:A363567 - cp's OEIS frontend

A363565 G.f. satisfies A(x) = exp( Sum_{k>=1} (2 * (-1)^k + A(x^k)) * x^k/k ).

1, -1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 2, 2, 1, 3, 5, 4, 4, 10, 13, 11, 16, 30, 34, 35, 58, 91, 99, 123, 200, 275, 315, 437, 671, 869, 1065, 1548, 2239, 2848, 3730, 5446, 7530, 9699, 13273, 19056, 25730, 33947, 47463, 66796, 89565, 120976, 170033, 235524

Offset: 0

Seiichi Manyama, Jun 10 2023

sign

Cf. A001678, A363566.

Cf. A363567.

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

A(x) = B(x)/(1 + x)^2 where B(x) is the g.f. of A363567.
A(x) = Sum_{k>=0} a(k) * x^k = 1/(1+x)^2 * 1/Product_{k>=0} (1-x^(k+1))^a(k).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 2 * (-1)^k + Sum_{d|k} d * a(d-1) ) * a(n-k).

A363575 G.f. satisfies A(x) = exp( Sum_{k>=1} A(x^k) * x^k/(k * (1 + x^k)^3) ).

Original entry on oeis.org

1, 1, -1, 1, 2, -4, -1, 10, -3, -20, 19, 38, -70, -65, 221, 73, -640, 117, 1745, -1223, -4433, 5770, 10124, -22007, -18999, 75063, 19307, -235725, 59665, 685744, -525477, -1832544, 2531982, 4364936, -10007555, -8468154, 35302510, 8542655, -114305453

Offset: 0

Seiichi Manyama, Jun 10 2023

sign

Cf. A198518, A363567.

Cf. A363548, A363566.

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

A(x) = (1 + x)^3 * B(x) where B(x) is the g.f. of A363566.

a(n) = Sum_{k=0..3} binomial(3,k) * A363566(n-k).

cp's OEIS Frontend

A363565 G.f. satisfies A(x) = exp( Sum_{k>=1} (2 * (-1)^k + A(x^k)) * x^k/k ).

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