A384470 -id:A384470 - cp's OEIS frontend

A384471 a(n) = Sum_{k=0..n} binomial(n,k)^2 * Stirling2(2k,k) Stirling2(2n-2k,n-k).

1, 2, 18, 306, 8046, 296100, 14307254, 865996306, 63308257198, 5432272670376, 535074966419260, 59461066810476232, 7354069129792197762, 1001371912804041913056, 148806933109572134044158, 23958722845801073318076450, 4154065510530807075869275150, 771608888261061026185781127184

Offset: 0

Vaclav Kotesovec, May 30 2025

nonn

Cf. A187655, A187657, A384470, A384472.

Cf. A226775.

Table[Sum[StirlingS2[2*k, k]*StirlingS2[2*n-2*k, n-k]*Binomial[n, k]^2, {k, 0, n}], {n, 0, 20}]

a(n) ~ 2^(3*n + 1/2) * n^(n - 3/2) / (Pi^(3/2) * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

A384472 a(n) = Sum_{k=0..n} binomial(n,k)^3 * Stirling2(2k,k) Stirling2(2n-2k,n-k).

Original entry on oeis.org

1, 2, 22, 558, 25506, 1770300, 166190354, 19647687682, 2798281247682, 466166725448544, 88942246964278060, 19127775950813311232, 4578817457796314714502, 1207681779462031251096888, 348018457509475159702959174, 108798555057988053563408904750, 36676526343321856806298038370210

Offset: 0

Vaclav Kotesovec, May 30 2025

nonn

In general, for m > 1, Sum_{k=0..n} binomial(n,k)^m * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) ~ 2^((m+1)*n + (m-1)/2) * n^(n-(m+1)/2) / (sqrt(m-1) * Pi^((m+1)/2) * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775.

Cf. A187655 (m=0), A187657 (m=1), A384471 (m=2), A384470.

Cf. A226775.

Table[Sum[StirlingS2[2*k, k]*StirlingS2[2*n-2*k, n-k]*Binomial[n, k]^3, {k, 0, n}], {n, 0, 20}]

a(n) ~ 2^(4*n + 1/2) * n^(n-2) / (Pi^2 * (1-w) * exp(n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

A384491 a(n) = n!^2 * Sum_{k=0..n} Stirling2(2k,k) Stirling2(2n-2k,n-k) / binomial(n,k)^2.

Original entry on oeis.org

1, 2, 57, 6536, 1966816, 1226860992, 1373652478656, 2507498281198080, 6966291361870181376, 27969794062091821670400, 155875927262331497576140800, 1167389777699203314381963264000, 11441270265465265986005655905894400, 143525982910350708912088976768630784000

Offset: 0

Vaclav Kotesovec, May 31 2025

nonn

Cf. A187655, A384470, A384492.

Cf. A187657, A384471, A384472.

Table[n!^2 * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k]^2, {k, 0, n}], {n, 0, 15}]

a(n) = n!^2 * sum(k=0, n, stirling(2*k,k, 2) * stirling(2*n-2*k,n-k,2) / binomial(n,k)^2); \\ Michel Marcus, May 31 2025

a(n) ~ sqrt(Pi) * 2^(2*n + 3/2) * n^(3*n + 1/2) / (sqrt(1-w) * exp(3*n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

A384492 a(n) = n!^3 * Sum_{k=0..n} Stirling2(2k,k) Stirling2(2n-2k,n-k) / binomial(n,k)^3.

Original entry on oeis.org

1, 2, 113, 38992, 47071264, 147015606528, 988250901343488, 12631667044878213120, 280790763724247161061376, 10147405862241529912885248000, 565550513462476798468573003776000, 46592777163703224212146175606784000000, 5479872142880875751798643810680954683392000

Offset: 0

Vaclav Kotesovec, May 31 2025

nonn

Cf. A187655, A384470, A384491.

Cf. A187657, A384471, A384472.

Table[n!^3 * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k]^3, {k, 0, n}], {n, 0, 15}]

a(n) = n!^3 * sum(k=0, n, stirling(2*k,k,2) * stirling(2*n-2*k,n-k,2) / binomial(n,k)^3); \\ Michel Marcus, May 31 2025

a(n) ~ Pi * 2^(2*n+2) * n^(4*n+1) / (sqrt(1-w) * exp(4*n) * (2-w)^n * w^n), where w = -LambertW(-2*exp(-2)) = -A226775 = 0.4063757399599599...

cp's OEIS Frontend

A384471 a(n) = Sum_{k=0..n} binomial(n,k)^2 * Stirling2(2k,k) Stirling2(2n-2k,n-k).

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A384472 a(n) = Sum_{k=0..n} binomial(n,k)^3 * Stirling2(2k,k) Stirling2(2n-2k,n-k).

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A384491 a(n) = n!^2 * Sum_{k=0..n} Stirling2(2k,k) Stirling2(2n-2k,n-k) / binomial(n,k)^2.

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A384492 a(n) = n!^3 * Sum_{k=0..n} Stirling2(2k,k) Stirling2(2n-2k,n-k) / binomial(n,k)^3.

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cp's OEIS Frontend

A384471 a(n) = Sum_{k=0..n} binomial(n,k)^2 * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k).

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A384472 a(n) = Sum_{k=0..n} binomial(n,k)^3 * Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k).

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A384491 a(n) = n!^2 * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k)^2.

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Formula

A384492 a(n) = n!^3 * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k)^3.

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Author

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Programs

Mathematica

PARI

Formula

A384471 a(n) = Sum_{k=0..n} binomial(n,k)^2 * Stirling2(2k,k) Stirling2(2n-2k,n-k).

A384472 a(n) = Sum_{k=0..n} binomial(n,k)^3 * Stirling2(2k,k) Stirling2(2n-2k,n-k).

A384491 a(n) = n!^2 * Sum_{k=0..n} Stirling2(2k,k) Stirling2(2n-2k,n-k) / binomial(n,k)^2.

A384492 a(n) = n!^3 * Sum_{k=0..n} Stirling2(2k,k) Stirling2(2n-2k,n-k) / binomial(n,k)^3.