A386300 -id:A386300 - cp's OEIS frontend

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

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).

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).

A386301 a(n) = 1 + Sum_{k=0..n-1} (-1)^k * a(k) * a(n-1-k).

Original entry on oeis.org

1, 2, 1, -1, 1, 8, 1, -28, 1, 134, 1, -649, 1, 3320, 1, -17497, 1, 94526, 1, -520507, 1, 2910896, 1, -16487794, 1, 94393106, 1, -545337199, 1, 3175320608, 1, -18615098836, 1, 109783526822, 1, -650884962907, 1, 3877184797784, 1, -23193307022860, 1, 139271612505362

Offset: 0

Seiichi Manyama, Jul 17 2025

sign

Cf. A007317, A348857, A348858, A348859.

Cf. A064641, A386300.

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

G.f. A(x) satisfies:
(1) A(x) = 1/( (1-x) * (1-x*A(-x)) ).
(2) A(x)*A(-x) = B(-x^2), where B(x) is the g.f. of A064641.
(3) A(x) = 1/(1-x) + 2*x/(1+x^2 + sqrt(1+6*x^2-3*x^4)).
a(2*n) = 1 and a(2*n+1) = 1 + (-1)^n * A064641(n) for n >= 0.

cp's OEIS Frontend

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

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

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Author

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Crossrefs

Programs

PARI

Formula

A386301 a(n) = 1 + Sum_{k=0..n-1} (-1)^k * a(k) * a(n-1-k).

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Programs

PARI

Formula