A355423 -id:A355423 - cp's OEIS frontend

A355421 Expansion of e.g.f. exp(Sum_{k=1..3} (exp(k*x) - 1)).

1, 6, 50, 504, 5870, 76872, 1111646, 17522664, 298133054, 5433157512, 105396184478, 2165189912040, 46901678992958, 1067332196912136, 25435754924426270, 633014456504059368, 16411191933603611198, 442258823578968351624

Offset: 0

Seiichi Manyama, Jul 01 2022

nonn

Column k=3 of A355423.

Cf. A004701, A306027, A355379, A355380.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, 3, exp(k*x)-1))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (1+2^j+3^j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} (1 + 2^k + 3^k) * binomial(n-1,k-1) * a(n-k).

A355422 Expansion of e.g.f. exp(Sum_{k=1..4} (exp(k*x) - 1)).

Original entry on oeis.org

1, 10, 130, 2000, 35054, 684000, 14628190, 338990000, 8438270014, 224070580800, 6311530677150, 187702155610000, 5870416574854974, 192423935736656800, 6591135679171866910, 235315671951948070000, 8736534653549465359934

Offset: 0

Seiichi Manyama, Jul 01 2022

nonn

Column k=4 of A355423.

Cf. A004702, A306028.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, 4, exp(k*x)-1))))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (1+2^j+3^j+4^j)*binomial(i-1, j-1)*v[i-j+1])); v;

a(0) = 1; a(n) = Sum_{k=1..n} (1 + 2^k + 3^k + 4^k) * binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

A355421 Expansion of e.g.f. exp(Sum_{k=1..3} (exp(k*x) - 1)).

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A355422 Expansion of e.g.f. exp(Sum_{k=1..4} (exp(k*x) - 1)).

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PARI

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Formula