A345104 -id:A345104 - cp's OEIS frontend

A345105 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

1, 4, 25, 247, 3283, 54661, 1092427, 25473037, 678837319, 20351864821, 677954261635, 24842157250117, 993040102321927, 43003754679356941, 2005536858420616963, 100211634039201328381, 5341144936822423446247, 302468060262966258380773, 18136282125753572653056355

Offset: 0

Ilya Gutkovskiy, Jun 08 2021

nonn

Cf. A032031, A054687, A345102, A345104.

a[n_] := a[n] = 1 + 3 Sum[Binomial[n - 1, k] a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; A[] = 1; Do[A[x] = Normal[Integrate[3 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) = 3 * A(x)^2 + exp(x).

A386298 a(n) = 1 + Sum_{k=0..n-1} 2^k * binomial(n-1,k) * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 7, 52, 721, 17594, 754063, 58139188, 8321310193, 2272187953346, 1206524396886823, 1260788083530821380, 2611061273843639666401, 10760136322351992470924570, 88437432027319862460463145551, 1451522912694521425631922482171812, 47608493474799808182534348919785356065

Offset: 0

Seiichi Manyama, Jul 17 2025

nonn

Cf. A054687, A386299, A386300.

Cf. A345104, A348857.

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

cp's OEIS Frontend

A345105 a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n-1,k) * a(k) * a(n-k-1).

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