A363510 -id:A363510 - cp's OEIS frontend

A363509 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (3 + A(x^k)) * x^k/k ).

1, 4, 10, 30, 101, 361, 1354, 5238, 20740, 83683, 342719, 1421019, 5953306, 25162342, 107163924, 459438524, 1981247950, 8588054014, 37398421941, 163534601567, 717776072291, 3161117717887, 13964782042188, 61866495037806, 274792382789958

Offset: 0

Seiichi Manyama, Jun 06 2023

nonn

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

Cf. A004111, A038075, A038076, A363510.

Cf. A363507.

terms = 25; A[] = 0; Do[A[x] = Exp[Sum[(-1)^(k+1)*(3+A[x^k])*x^k/k,{k,terms}]]+ O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, May 10 2025 *)

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

A(x) = Sum_{k>=0} a(k) * x^k = (1+x)^3 * Product_{k>=0} (1+x^(k+1))^a(k).

a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( 3 * (-1)^k + Sum_{d|k} (-1)^(k/d) * d * a(d-1) ) * a(n-k).

A363508 G.f. satisfies A(x) = exp( Sum_{k>=1} (4 + A(x^k)) * x^k/k ).

Original entry on oeis.org

1, 5, 20, 80, 340, 1516, 7046, 33736, 165436, 826566, 4193348, 21542664, 111848161, 585949358, 3093526496, 16442687695, 87914559018, 472522551440, 2551591234444, 13836226412386, 75311992329508, 411336641019998, 2253641429297336

Offset: 0

Seiichi Manyama, Jun 06 2023

nonn

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

Cf. A000081, A029857, A036249, A363507.

Cf. A363510.

seq(n) = my(A=1); for(i=1, n, A=exp(sum(k=1, i, (4+subst(A, x, x^k))*x^k/k)+x*O(x^n))); Vec(A);

A(x) = Sum_{k>=0} a(k) * x^k = 1/(1-x)^4 * 1/Product_{k>=0} (1-x^(k+1))^a(k).

a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( 4 + Sum_{d|k} d * a(d-1) ) * a(n-k).

cp's OEIS Frontend

A363509 G.f. satisfies A(x) = exp( Sum_{k>=1} (-1)^(k+1) * (3 + A(x^k)) * x^k/k ).

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