A377098 -id:A377098 - cp's OEIS frontend

A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(nk,k) / ((n-1)k+1).

1, 1, 3, 19, 219, 3901, 95838, 3022909, 116798643, 5350403737, 283728025998, 17104314563843, 1155635807408096, 86513627563199279, 7109252862969177287, 636268582522962837475, 61610670571434193189443, 6418044336586421956746033, 715718717341021991299583730

Offset: 0

Vaclav Kotesovec, Nov 23 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..338

Cf. A002212, A307678, A349331, A349332, A349333, A349334, A349335.

Cf. A377098, A378325, A378327.

Table[Sum[Binomial[n-1, k-1]*Binomial[n*k, k]/((n-1)*k+1), {k, 0, n}], {n, 0, 20}]

a(n) ~ exp(n + exp(-1) - 1/2) * n^(n - 5/2) / sqrt(2*Pi).

A378325 G.f. A(x) = Sum_{n>=0} a(n)*x^n, where a(n) = Sum_{k=0..n-1} [x^k] A(x)^k for n >= 1 with a(0) = 1.

Original entry on oeis.org

1, 1, 2, 7, 41, 338, 3499, 42969, 606351, 9633640, 169888025, 3290314970, 69409429043, 1584105116525, 38894316619948, 1022411500472240, 28653072049382809, 852911635849385778, 26876978490909421289, 893929164892155754432, 31296785296935394097351, 1150551256823546563078988

Offset: 0

Vaclav Kotesovec, Nov 23 2024

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A377098, A378326.

Cf. A002212, A307678, A349331, A349332, A349333, A349334, A349335.

{a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0);
A[#A] = 1 + sum(k=1, m-1, (polcoeff(Ser(A)^k, k)) )); A[n+1]}
for(n=0, 30, print1(a(n), ", ")) \\ after Paul D. Hanna

a(n) ~ c * n! / (n^alpha * LambertW(1)^n), where alpha = 2 - 2*LambertW(1) - 1/(1 + LambertW(1)) = 0.22760967581532... and c = 0.323194722450152336...

cp's OEIS Frontend

A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(nk,k) / ((n-1)k+1).

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Formula

A378325 G.f. A(x) = Sum_{n>=0} a(n)*x^n, where a(n) = Sum_{k=0..n-1} [x^k] A(x)^k for n >= 1 with a(0) = 1.

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PARI

Formula

cp's OEIS Frontend

A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(n*k,k) / ((n-1)*k+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A378325 G.f. A(x) = Sum_{n>=0} a(n)*x^n, where a(n) = Sum_{k=0..n-1} [x^k] A(x)^k for n >= 1 with a(0) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A378326 a(n) = Sum_{k=0..n} binomial(n-1,k-1) * binomial(nk,k) / ((n-1)k+1).