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%I #19 Dec 01 2021 13:00:51
%S 1,1,3,7,49,201,1411,7183,68097,453169,4523491,34273911,403454833,
%T 3618761017,45157828899,445900023871,6206361667201,69111310499553,
%U 1017103374816067,12237616620289639,195222691795726641,2575612811875082281,43240905591424459843,608870179599833137647
%N E.g.f. exp(x+x^2+x^4).
%F E.g.f. exp(x+x^2+x^4).
%F a(n)=n!*sum(k=1..n, sum(j=floor((4*k-n)/3)..floor((4*k-n)/2), binomial(j,n-4*k+3*j)*binomial(k,j))/k!), n>0, a(0)=1.
%F D-finite with recurrence a(n) = a(n-1) + 2*(n-1)*a(n-2) + 4*(n-3)*(n-2)*(n-1)*a(n-4). - _Vaclav Kotesovec_, Oct 09 2013
%F a(n) ~ 2^(n/2-1) * n^(3*n/4) * exp(n^(1/4)/sqrt(2) - 3*n/4 + sqrt(n)/2 - 1/8) * (1 - 1/(4*sqrt(2)*n^(1/4)) + 43/(192*sqrt(n)) + 271/(768*sqrt(2)*n^(3/4))). - _Vaclav Kotesovec_, Oct 09 2013
%t CoefficientList[Series[E^(x+x^2+x^4), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 09 2013 *)
%o (Maxima)
%o a(n):=n!*sum(sum(binomial(j,n-4*k+3*j)*binomial(k,j),j,floor((4*k-n)/3),floor((4*k-n)/2))/k!,k,1,n);
%o (PARI)
%o N=33; x='x+O('x^N);
%o egf=exp(x+x^2+x^4);
%o Vec(serlaplace(egf))
%o /* _Joerg Arndt_, Sep 15 2012 */
%K nonn
%O 0,3
%A _Vladimir Kruchinin_, May 24 2011
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