cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A187657 Binomial convolution of the central Stirling numbers of the second kind.

Original entry on oeis.org

1, 2, 16, 222, 4416, 114660, 3676814, 140408338, 6222858240, 314006546124, 17774855765140, 1115507717954432, 76871991664546170, 5770732305836768712, 468750121409142448386, 40964179307489016777630, 3832326196169482368117760
Offset: 0

Views

Author

Emanuele Munarini, Mar 12 2011

Keywords

Crossrefs

Cf. A384471, A384472.

Programs

  • Maple
    seq(sum(binomial(n, k) *combinat[stirling2](2*k, k) *combinat[stirling2](2*(n-k), n-k), k=0..n), n=0..12);
  • Mathematica
    Table[Sum[Binomial[n, k]StirlingS2[2k, k]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 16}]
  • Maxima
    makelist(sum(binomial(n,k)*stirling2(2*k,k)*stirling2(2*n-2*k, n-k),k,0,n),n,0,12);

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * S(2k,k) * S(2n-2k,n-k).
Limit n->infinity (a(n)/n!)^(1/n) = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.17655460948348... . - Vaclav Kotesovec, Jun 01 2015

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

Views

Author

Vaclav Kotesovec, May 30 2025

Keywords

Comments

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.

Crossrefs

Cf. A187655 (m=0), A187657 (m=1), A384471 (m=2), A384470.
Cf. A226775.

Programs

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

Formula

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...

A384470 a(n) = n! * Sum_{k=0..n} Stirling2(2*k,k) * Stirling2(2*n-2*k,n-k) / binomial(n,k).

Original entry on oeis.org

1, 2, 29, 1108, 82924, 10302768, 1917699552, 499332175200, 173242955039616, 77238974345915520, 43027312823342164800, 29285800226400628915200, 23913110797474508388449280, 23071378298963178620672409600, 25964692904608781751347296204800, 33711625062334209438536728660070400
Offset: 0

Views

Author

Vaclav Kotesovec, May 30 2025

Keywords

Crossrefs

Cf. A187655, A187657, A384471, A384472.
Cf. A226775.

Programs

  • Mathematica
    Table[n! * Sum[StirlingS2[2*k, k] * StirlingS2[2*n-2*k, n-k] / Binomial[n, k], {k, 0, n}], {n, 0, 20}]

Formula

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

A384495 a(n) = Sum_{k=0..n} binomial(n,k)^2 * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)).

Original entry on oeis.org

1, 2, 26, 648, 25094, 1372100, 99827020, 9233563136, 1045169591270, 140259346792380, 21754963505429340, 3823376222328582480, 749784319125445476092, 162122841942093462239368, 38288723630416561023861048, 9801732906198391239249940800, 2702731846233390353066363949830
Offset: 0

Views

Author

Vaclav Kotesovec, May 31 2025

Keywords

Crossrefs

Cf. A187656, A187658, A384496, A384471.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]^2 * Abs[StirlingS1[2*k,k]] * Abs[StirlingS1[2*n-2*k, n-k]], {k, 0, n}], {n, 0, 20}]

Formula

a(n) ~ 2^(4*n - 1/2) * n^(n - 3/2) * w^(2*n) / ((w-1) * Pi^(3/2) * exp(n) * (2*w-1)^n), where w = -LambertW(-1, -exp(-1/2)/2) = 1.7564312086261696769827376166...

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

Views

Author

Vaclav Kotesovec, May 31 2025

Keywords

Crossrefs

Cf. A187655, A384470, A384492.
Cf. A187657, A384471, A384472.

Programs

  • Mathematica
    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}]
  • PARI
    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

Formula

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(2*k,k) * Stirling2(2*n-2*k,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

Views

Author

Vaclav Kotesovec, May 31 2025

Keywords

Crossrefs

Cf. A187655, A384470, A384491.
Cf. A187657, A384471, A384472.

Programs

  • Mathematica
    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}]
  • PARI
    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

Formula

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...
Showing 1-6 of 6 results.

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