A342111 a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n,k) * Stirling1(n,n-k).

1, 0, 1, 12, 193, 3980, 100805, 3034920, 105994833, 4215106728, 188097696345, 9309515255700, 506149663220641, 29989851619249236, 1923467938147053389, 132771455705186298000, 9814431285244231295265, 773520674985391641371280, 64752473306596841023424945

Offset

0,4

Links

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

Formula

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

d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.9108149645682558987515348052403521978987052817678471761394112...

c = 1/(4*sqrt(-LambertW(-1, -exp(-1/2)/2)) * sqrt(-1 - LambertW(-1, -exp(-1/2)/2))*Pi) = 0.06903826111269387517867145566264007373042059749428879149076344304196548... - Vaclav Kotesovec, Feb 28 2021, updated May 14 2025

a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^2. - Seiichi Manyama, May 13 2025

Mathematica

Table[(-1)^n*Sum[StirlingS1[n, k]*StirlingS1[n, n-k], {k, 0, n}], {n, 0, 20}]

Prog

(PARI) a(n) = (-1)^n*sum(k=0, n, stirling(n, k, 1)*stirling(n, n-k, 1)); \\ Michel Marcus, Feb 28 2021
(Magma) [(&+[(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 03 2021
(SageMath) [sum( stirling_number1(n, k)*stirling_number1(n, n-k) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Jun 03 2021

Crossrefs

Cf. A008275, A014322, A047796, A132393, A342110.

Cf. A048994, A155826.

Cf. A129256, A187235, A246117.

Sequence in context: A328261 A264603 A296841 * A296942 A397058 A159498

Adjacent sequences: A342108 A342109 A342110 * A342112 A342113 A342114

Keyword

nonn

Author

Vaclav Kotesovec, Feb 28 2021

Status

approved