A375689 -id:A375689 - cp's OEIS frontend

A375688 Expansion of e.g.f. 1 / (1 + 3 * x * log(1 - x))^(1/3).

1, 0, 2, 3, 56, 270, 4824, 44520, 866816, 12195792, 267873120, 5073187680, 126754229568, 2999710359360, 85061489235072, 2400155295632640, 76724104598031360, 2502434971473937920, 89428428468644493312, 3300036525511418327040

Offset: 0

Seiichi Manyama, Aug 24 2024

nonn

Cf. A367879, A375689.

Cf. A347015.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*x*log(1-x))^(1/3)))

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

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

A375697 Expansion of e.g.f. 1 / (1 - 3 * x * (exp(x) - 1))^(2/3).

Original entry on oeis.org

1, 0, 4, 6, 128, 610, 11712, 107114, 2167776, 30285378, 678296720, 12761459722, 321364284144, 7550564959106, 214210299545088, 5993932335381930, 190756625697021632, 6161493279498219394, 218469987108304908336, 7972839360644407925258

Offset: 0

Seiichi Manyama, Aug 24 2024

nonn

Cf. A367881, A375696.

Cf. A375689.

my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*(exp(x)-1))^(2/3)))

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

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

cp's OEIS Frontend

A375688 Expansion of e.g.f. 1 / (1 + 3 * x * log(1 - x))^(1/3).

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A375697 Expansion of e.g.f. 1 / (1 - 3 * x * (exp(x) - 1))^(2/3).

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Author

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Crossrefs

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PARI

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Formula