A363543 -id:A363543 - cp's OEIS frontend

A363541 G.f. satisfies A(x) = exp( Sum_{k>=1} (3^k + A(x^k)) * x^k/k ).

1, 4, 17, 73, 324, 1469, 6838, 32490, 157398, 775010, 3870690, 19567202, 99957231, 515250057, 2676884745, 14002926871, 73693381322, 389904743248, 2072794614996, 11066421965311, 59310040841395, 318978744562253, 1720962766007827

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

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

Cf. A001678, A036249, A362389.

Cf. A363507, A363543, A363546.

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 A363546.
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).

A363542 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (2^k + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 3, 5, 14, 38, 114, 360, 1166, 3872, 13094, 44961, 156244, 548636, 1943333, 6935817, 24917586, 90039163, 327029681, 1193258619, 4371901789, 16077606949, 59325057056, 219579151797, 815017718383, 3032959638204, 11313632991360, 42295634914403

Offset: 0

Seiichi Manyama, Jun 09 2023

nonn

Cf. A038075, A363543.

Cf. A362389.

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

A(x) = Sum_{k>=0} a(k) * x^k = (1+2*x) * Product_{k>=0} (1+x^(k+1))^a(k).

a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( (-2)^k + Sum_{d|k} (-1)^(k/d) * d * a(d-1) ) * a(n-k).

cp's OEIS Frontend

A363541 G.f. satisfies A(x) = exp( Sum_{k>=1} (3^k + A(x^k)) * x^k/k ).

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