A375716 -id:A375716 - cp's OEIS frontend

A375718 Expansion of e.g.f. 1 / sqrt(1 - x^3 * (exp(x) - 1)).

1, 0, 0, 0, 12, 30, 60, 105, 15288, 136332, 794160, 3742695, 165156420, 2977295178, 34259966832, 307175369865, 8066201665200, 210501545175960, 3893163654156768, 56023707973290507, 1275541469736173820, 38629328708426716470, 991445561747177496960

Offset: 0

Seiichi Manyama, Aug 25 2024

nonn

Cf. A358014, A375716, A375717.

Cf. A375701.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1-x^3*(exp(x)-1))))

a(n) = n!*sum(k=0, n\4, prod(j=0, k-1, 6*j+3)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+3)) * Stirling2(n-3*k,k)/(6^k*(n-3k)!).

A375717 Expansion of e.g.f. 1 / (1 - x^3 * (exp(x) - 1))^(1/3).

Original entry on oeis.org

1, 0, 0, 0, 8, 20, 40, 70, 9072, 80808, 470640, 2217930, 91956920, 1649007932, 18956858648, 169921752910, 4310715370080, 111302746115920, 2053356893604192, 29525879498171538, 660295352236840680, 19735183465373056100, 504257138580203577800

Offset: 0

Seiichi Manyama, Aug 25 2024

nonn

Cf. A358014, A375716, A375718.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^3*(exp(x)-1))^(1/3)))

a(n) = n!*sum(k=0, n\4, prod(j=0, k-1, 6*j+2)*stirling(n-3*k, k, 2)/(6^k*(n-3*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+2)) * Stirling2(n-3*k,k)/(6^k*(n-3k)!).

cp's OEIS Frontend

A375718 Expansion of e.g.f. 1 / sqrt(1 - x^3 * (exp(x) - 1)).

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A375717 Expansion of e.g.f. 1 / (1 - x^3 * (exp(x) - 1))^(1/3).

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