A345105 -id:A345105 - cp's OEIS frontend

A345104 a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

1, 3, 13, 89, 825, 9601, 134185, 2188353, 40788745, 855303265, 19927758377, 510728051073, 14279388168137, 432505475357729, 14107767947949289, 493046896702987841, 18380057918926012809, 728005164671113691105, 30531323352522247757225, 1351567976217998536472833

Offset: 0

Ilya Gutkovskiy, Jun 08 2021

nonn

Cf. A000165, A052886, A054687, A345105.

a[n_] := a[n] = 1 + 2 Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; A[] = 1; Do[A[x] = Normal[Integrate[2 A[x]^2 + Exp[x], x] + O[x]^(nmax + 1)], nmax]; CoefficientList[A[x], x] Range[0, nmax]!

E.g.f. A(x) satisfies: A'(x) = 2 * A(x)^2 + exp(x).

A386299 a(n) = 1 + Sum_{k=0..n-1} 3^k * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 9, 115, 3869, 349233, 88835413, 65934937157, 145194342935565, 955092851917410169, 18817250316042492760133, 1111535058740789497290819885, 196930668231818953760620540315069, 104661954649505883286587026252584631249, 166867787421063078832424708621648215185207669

Offset: 0

Seiichi Manyama, Jul 17 2025

nonn

Cf. A054687, A386298, A386300.

Cf. A345105, A348858.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=1+sum(j=0, i-1, 3^j*binomial(i-1, j)*v[j+1]*v[i-j])); v;

E.g.f. A(x) satisfies A'(x) = exp(x) + A(x) * A(3*x).

cp's OEIS Frontend

A345104 a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

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