A187652 - OEIS

%I #12 Oct 19 2024 08:33:20

%S 1,0,10,194,5932,237624,11820780,702992968,48662470640,3843811669088,

%T 341207224961856,33627579102171680,3643463136559851440,

%U 430456189350273371648,55075003474909952394848,7586546772496980353804704

%N Alternated binomial cumulative sums of the (signless) central Stirling numbers of the first kind (A187646).

%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*s(2*k,k).

%F a(n) ~ c * d^n * (n-1)!, where d = 8*w^2/(2*w-1), where w = -LambertW(-1,-exp(-1/2)/2) = 1.7564312086261696769827376166... and c = 0.11686978539934159049334861225275481804523808136863346883911376048... - _Vaclav Kotesovec_, Jul 05 2021

%p seq(sum((-1)^(n-k)*binomial(n,k)*abs(combinat[stirling1](2*k,k)),k=0..n),n=0..12);

%t Table[Sum[(-1)^(n - k)Binomial[n, k]Abs[StirlingS1[2k, k]], {k, 0, n}], {n, 0, 15}]

%o (Maxima) makelist(sum((-1)^(n-k)*binomial(n,k)*abs(stirling1(2*k,k)),k,0,n),n,0,12);

%Y Cf. A187646.

%K nonn,easy

%O 0,3

%A _Emanuele Munarini_, Mar 12 2011