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%I #16 Feb 10 2024 04:44:52
%S 1,3,21,234,3590,70254,1672972,46955760,1517994792,55549351800,
%T 2269918543640,102452561694864,5062050729973120,271751784988056576,
%U 15750949414628405760,980315266648197537792,65207656047198387921536
%N a(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+k)^(n-1).
%F E.g.f.: A(x) = log(B(x)), where B(x) is e.g.f. of A138860.
%F E.g.f.: A(x) = Series_Reversion[ 2*x/(exp(x) + exp(2*x)) ].
%F a(n) ~ n^(n-1)*(1+r)^n*r^n/(sqrt(1+3*r)*(1-r)^(2*n)*exp(n)*2^n), where r = 0.6472709258412625... is the root of the equation (r/(1-r))^(1+r) = e. - _Vaclav Kotesovec_, Jun 15 2013
%p A138903 := proc(n) local k ; add(binomial(n,k)*(n+k)^(n-1),k=0..n)/2^n ; end: seq(A138903(n),n=1..20) ; # _R. J. Mathar_, Apr 12 2008
%t Table[1/2^n * Sum[Binomial[n,k]*(n+k)^(n-1),{k,0,n}],{n,1,20}] (* _Vaclav Kotesovec_, Jun 15 2013 *)
%o (PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(serreverse(2*x/(exp(X)+exp(2*X)) ), n)}
%K easy,nonn
%O 1,2
%A _Paul D. Hanna_ and _Vladeta Jovovic_, Apr 02 2008, Apr 03 2008
%E More terms from _R. J. Mathar_, Apr 12 2008
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