A384495 -id:A384495 - cp's OEIS frontend

A187658 Binomial convolution of the (signless) central Stirling numbers of the first kind (A187646).

1, 2, 24, 516, 16064, 655840, 33157240, 1999679696, 140128848384, 11189643689088, 1003005057594240, 99725721676986240, 10892178742891589792, 1296379044138734510656, 166999512859041432577280, 23149972436862049305233280

Offset: 0

Emanuele Munarini, Mar 12 2011

Cf. A187646, A384495, A384496.

seq(sum(binomial(n,k)*abs(combinat[stirling1](2*k,k))*abs(combinat[stirling1](2*(n-k),n-k)),k=0..n),n=0..12);

Table[Sum[Binomial[n, k]Abs[StirlingS1[2k, k]]Abs[StirlingS1[2n - 2k, n - k]], {k, 0, n}], {n, 0, 15}]

makelist(sum(binomial(n,k)*abs(stirling1(2*k,k))*abs(stirling1(2*n-2*k,n-k)),k,0,n),n,0,12);

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

Limit_{n->oo} (a(n)/n!)^(1/n) = -8*LambertW(-1, -exp(-1/2)/2)^2 / (1 + 2*LambertW(-1, -exp(-1/2)/2)) = 9.821629929136511797503... - Vaclav Kotesovec, May 30 2025

A384496 a(n) = Sum_{k=0..n} binomial(n,k)^3 * abs(Stirling1(2k,k)) abs(Stirling1(2n-2k,n-k)).

Original entry on oeis.org

1, 2, 30, 1044, 68474, 7180900, 1050625720, 196205015216, 44361477901818, 11751610490415828, 3567182462164189140, 1220655384720089761080, 464932034143270233958352, 195108754505934104188716064, 89452431045403310104416682304, 44489455448017524780072427344000

Offset: 0

Vaclav Kotesovec, May 31 2025

nonn

In general, for m > 1, Sum_{k=0..n} binomial(n,k)^m * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)) ~ 2^((m+2)*n + (m-3)/2) * n^(n - (m+1)/2) * w^(2*n) / (sqrt(m-1) * (w-1) * Pi^((m+1)/2) * exp(n) * (2*w-1)^n), where w = -LambertW(-1, -exp(-1/2)/2).

Cf. A187656 (m=0), A187658 (m=1), A384495 (m=2), A384472.

Table[Sum[Binomial[n,k]^3 * Abs[StirlingS1[2*k,k]] * Abs[StirlingS1[2*n-2*k,n-k]], {k, 0, n}], {n, 0, 20}]

a(n) ~ 2^(5*n - 1/2) * n^(n-2) * w^(2*n) / ((w-1) * Pi^2 * exp(n) * (2*w-1)^n), where w = -LambertW(-1, -exp(-1/2)/2) = 1.7564312086261696769827376166...

cp's OEIS Frontend

A187658 Binomial convolution of the (signless) central Stirling numbers of the first kind (A187646).

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Formula

A384496 a(n) = Sum_{k=0..n} binomial(n,k)^3 * abs(Stirling1(2k,k)) abs(Stirling1(2n-2k,n-k)).

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cp's OEIS Frontend

A187658 Binomial convolution of the (signless) central Stirling numbers of the first kind (A187646).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

A384496 a(n) = Sum_{k=0..n} binomial(n,k)^3 * abs(Stirling1(2*k,k)) * abs(Stirling1(2*n-2*k,n-k)).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A384496 a(n) = Sum_{k=0..n} binomial(n,k)^3 * abs(Stirling1(2k,k)) abs(Stirling1(2n-2k,n-k)).