A375699 -id:A375699 - cp's OEIS frontend

A375701 Expansion of e.g.f. 1 / sqrt(1 + x^3 * log(1 - x)).

1, 0, 0, 0, 12, 30, 120, 630, 19152, 166320, 1506600, 14968800, 313014240, 4864860000, 72829607760, 1116874558800, 23605893400320, 495461472105600, 10289649464640000, 215706738207542400, 5222625647551920000, 133507746422859513600, 3481696859911699968000

Offset: 0

Seiichi Manyama, Aug 25 2024

nonn

Cf. A351504, A375699, A375700.

Cf. A375718.

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

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

a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+3)) * |Stirling1(n-3*k,k)|/(6^k*(n-3k)!).

A375700 Expansion of e.g.f. 1 / (1 + x^3 * log(1 - x))^(1/3).

Original entry on oeis.org

1, 0, 0, 0, 8, 20, 80, 420, 11648, 100800, 912000, 9055200, 181547520, 2790627840, 41568334080, 635617382400, 13172198645760, 273158953267200, 5632405756723200, 117530452124467200, 2815021136030515200, 71252240659839590400, 1844362570865444044800

Offset: 0

Seiichi Manyama, Aug 25 2024

nonn

Cf. A351504, A375699, A375701.

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

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

a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+2)) * |Stirling1(n-3*k,k)|/(6^k*(n-3k)!).

cp's OEIS Frontend

A375701 Expansion of e.g.f. 1 / sqrt(1 + x^3 * log(1 - x)).

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A375700 Expansion of e.g.f. 1 / (1 + x^3 * log(1 - x))^(1/3).

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Author

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PARI

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Formula