A202832 - OEIS

%I #14 Oct 07 2024 20:27:19

%S 1,2,9,38,211,1182,7639,50738,368841,2767202,22132249,182624598,

%T 1582522891,14122521662,131109031239,1250794578818,12334766500561,

%U 124733099306562,1297921351160809,13821821639912198,150946171640101251,1684074507271422302,19217497036753475959

%N E.g.f: exp(2*x + 5*x^2/2).

%H Vincenzo Librandi, <a href="/A202832/b202832.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = Sum_{k=0..[n/2]} 2^(n-3*k)*5^k * n!/((n-2*k)!*k!).

%F O.g.f.: 1/(1-2*x - 5*x^2/(1-2*x - 10*x^2/(1-2*x - 15*x^2/(1-2*x - 20*x^2/(1-2*x -...))))), a continued fraction.

%F Recurrence: a(n) = 2*a(n-1) + 5*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 20 2012

%F a(n) ~ exp(2/5*sqrt(5*n)-n/2-1/5)*5^(n/2)*n^(n/2)/sqrt(2)*(1+17/150*sqrt(5)/sqrt(n)). - _Vaclav Kotesovec_, Oct 20 2012

%e E.g.f.: 1 + 2*x + 9*x^2/2! + 38*x^3/3! + 211*x^4/4! + 1182*x^5/5! +...

%t CoefficientList[Series[E^(2*x+5*x^2/2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 20 2012 *)

%o (PARI) {a(n)=n!*polcoeff(exp(2*x+5*x^2/2+x*O(x^n)),n)}

%o (PARI) {a(n)=sum(k=0,n\2,2^(n-3*k)*5^k*n!/((n-2*k)!*k!))}

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

%o {a(n)=local(CF=1+2*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-2*x-5*(n-k)*x^2*CF)); polcoeff(CF, n)}

%Y Column k=5 of A376826.

%Y Cf. A202831.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 25 2011