A342110 a(n) = Sum_{k=0..n} Stirling2(n,k) * Stirling2(n,n-k).

1, 0, 1, 6, 61, 770, 12160, 228382, 4989621, 124262532, 3475892685, 107901412520, 3681266754660, 136918473752216, 5513911474915116, 239034083286873630, 11098790133822288645, 549539910028075555016, 28903562131933534643851, 1609321474965547356327246

Offset: 0

Author

Vaclav Kotesovec, Feb 28 2021

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..350

Crossrefs

Cf. A008277, A014322, A047797, A342111.

Cf. A048993.

Programs

Magma

[(&+[StirlingSecond(n, k)*StirlingSecond(n, n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 03 2021

Mathematica

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

PARI

a(n) = sum(k=0, n, stirling(n, k, 2)*stirling(n, n-k, 2)); \\ Michel Marcus, Feb 28 2021

Sage

[sum( stirling_number2(n, k)*stirling_number2(n, n-k) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Jun 03 2021

Formula

From Vaclav Kotesovec, Feb 28 2021, updated May 25 2025: (Start)

a(n) ~ c * d^n * (n-1)!, where

d = A238258 = -2 / (LambertW(-2*exp(-2)) * (2 + LambertW(-2*exp(-2)))) = 3.0882773047417401791158400820254382768364448971420138767247...

c = 1/(2*Pi*sqrt((1 + LambertW(-2*exp(-2)))*(3 + LambertW(-2*exp(-2))))) = 0.12826577250734152801558828593238744179869387423941684693208180123477... (End)