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Alternating sums of the numbers in sequence A080253.
5

%I #11 Nov 27 2017 06:20:41

%S 1,2,15,132,1565,22918,400939,8160008,189453369,4942271754,

%T 143128015943,4556517918604,158167223290453,5945611873120910,

%U 240619359452963427,10430922482219093520,482234053313600047217,23683786738296923795986

%N Alternating sums of the numbers in sequence A080253.

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

%F 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

%F a(n) ~ n! * 2^(n - 1/2) / (log(2))^(n+1). - _Vaclav Kotesovec_, Nov 27 2017

%t 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}]

%t 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 *)

%o (Maxima) t(n):=sum(stirling2(n,k)*k!,k,0,n);

%o c(n):=sum(binomial(n,k)*2^k*t(k),k,0,n);

%o makelist(sum((-1)^(n-k)*c(k),k,0,n),n,0,10);

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

%K nonn

%O 0,2

%A _Emanuele Munarini_, Oct 04 2012