A385845 -id:A385845 - cp's OEIS frontend

A385844 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x^3*A''(x))).

1, 1, 1, 3, 21, 273, 5737, 177919, 7651849, 436186313, 31842549569, 2897710853939, 321648004495773, 42779331295225353, 6716367934603667145, 1229096733282700520799, 259339594018913458094865, 62500870590534491566841265, 17062742827503910747790541249, 5238263128497776755775631825219

Offset: 0

Seiichi Manyama, Jul 09 2025

nonn

Cf. A143917, A385845, A385846.

Cf. A385758, A385762.

terms = 20; A[] = 0; Do[A[x] = 1/((1 - x) * (1 - x^3*A''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]  (* Stefano Spezia, Jul 10 2025 *)

a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, sum(k=1, 2, stirling(2, k, 1)*j^k)*v[j+1]*v[i-j])); v;

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

A385846 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x^5*A''''(x))).

Original entry on oeis.org

1, 1, 1, 1, 1, 25, 3025, 1092025, 918393025, 1543818675025, 4670051491951201, 23541729570926148241, 186474039931306081488961, 2215498068423847604734793641, 38020162352221648825602734209201, 913434400512125113270449340963296649, 29925024395177730837015182640209851847809

Offset: 0

Seiichi Manyama, Jul 09 2025

nonn

Cf. A143917, A385844, A385845.

Cf. A385760.

terms = 17; A[] = 0; Do[A[x] = 1/((1 - x) * (1 - x^5*A''''[x])) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]  (* Stefano Spezia, Jul 10 2025 *)

a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=1+sum(j=0, i-1, sum(k=1, 4, stirling(4, k, 1)*j^k)*v[j+1]*v[i-j])); v;

a(n) = 1 + Sum_{k=0..n-1} (-6*k + 11*k^2 - 6*k^3 + k^4) * a(k) * a(n-1-k).

cp's OEIS Frontend

A385844 G.f. A(x) satisfies A(x) = 1/((1 - x) * (1 - x^3*A''(x))).

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