A217483 Alternating sums of the numbers in sequence A080253.

1, 2, 15, 132, 1565, 22918, 400939, 8160008, 189453369, 4942271754, 143128015943, 4556517918604, 158167223290453, 5945611873120910, 240619359452963427, 10430922482219093520, 482234053313600047217, 23683786738296923795986

Offset: 0

Author

Emanuele Munarini, Oct 04 2012

Keywords

• nonn

Crossrefs

Cf. A080253, A000670, A217484, A217485, A217486, A217487, A217488.

Programs

Mathematica

t[n_] := Sum[StirlingS2[n, k] k!, {k, 0, n}]; c[n_] := Sum[Binomial[n, k] 2^k t[k], {k, 0, n}]; Table[Sum[(-1)^(n-k)c[k],{k,0,n}], {n, 0, 100}]
nmax = 20; CoefficientList[Series[E^x/(2 - E^(2*x)) + Log[2 - E^(2*x)] / (2*E^x), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 27 2017 *)

Maxima

t(n):=sum(stirling2(n,k)*k!,k,0,n);
c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);
makelist(sum((-1)^(n-k)*c(k),k,0,n),n,0,10);

Formula

a(n) = sum((-1)^(n-k)*c(k),k=0..n), where c(n) = A080253(n).

E.g.f.: exp(x)/(2-exp(2*x)) - (1/2)*exp(-x)*log(1/(2-exp(2*x))). - corrected by Vaclav Kotesovec, Nov 27 2017

a(n) ~ n! * 2^(n - 1/2) / (log(2))^(n+1). - Vaclav Kotesovec, Nov 27 2017