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E.g.f.: exp(4*x + x^2/2).
3

%I #8 Oct 21 2012 00:01:06

%S 1,4,17,76,355,1724,8671,45028,240809,1323460,7461121,43079084,

%T 254388667,1534503676,9445067375,59263320964,378729294481,

%U 2463130313348,16290919259569,109500022678540,747527556645971,5180110680154684,36418521410184127,259636520604139556

%N E.g.f.: exp(4*x + x^2/2).

%F a(n) = Sum_{k=0..[n/2]} 4^(n-2*k)/2^k * n!/((n-2*k)!*k!).

%F O.g.f.: 1/(1-4*x - x^2/(1-4*x - 2*x^2/(1-4*x - 3*x^2/(1-4*x - 4*x^2/(1-4*x -...))))), a continued fraction.

%F Recurrence: a(n) = 4*a(n-1) + (n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 20 2012

%F a(n) ~ exp(4*sqrt(n)-n/2-4)*n^(n/2)/sqrt(2)*(1+11/(3*sqrt(n))). - _Vaclav Kotesovec_, Oct 20 2012

%e E.g.f.: A(x) = 1 + 4*x + 17*x^2/2! + 76*x^3/3! + 355*x^4/4! + 1724*x^5/5! +...

%t CoefficientList[Series[E^(4*x+x^2/2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 20 2012 *)

%o (PARI) {a(n)=n!*polcoeff(exp(4*x+x^2/2+x*O(x^n)),n)}

%o (PARI) {a(n)=sum(k=0,n\2,4^(n-2*k)/2^k * n!/((n-2*k)!*k!))}

%o (PARI) /* O.g.f. as a continued fraction: */

%o {a(n)=local(CF=1+4*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-4*x-(n-k)*x^2*CF)); polcoeff(CF, n)}

%Y Cf. A202878.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 25 2011