A202834 - OEIS

%I #17 Dec 24 2020 02:33:13

%S 1,3,10,36,138,558,2364,10440,47868,227124,1112184,5607792,29057400,

%T 154465704,841143312,4685949792,26674999056,155000193840,918475565472,

%U 5545430185536,34087326300576,213170582612448,1355345600149440,8755789617922176,57440317657203648

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

%H Vincenzo Librandi, <a href="/A202834/b202834.txt">Table of n, a(n) for n = 0..200</a>

%H Yasushi Ieno, <a href="https://arxiv.org/abs/2012.12655">A newly-generalized problem from a problem for the Mathematical Olympiad and the methods to solve it</a>, arXiv:2012.12655 [math.GM], 2020. See p. 9.

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

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

%F a(n) ~ n^(n/2)*exp(-n/2+3*sqrt(n)-9/4)/sqrt(2) * (1+15/(8*sqrt(n))). - _Vaclav Kotesovec_, May 23 2013

%F Recurrence: a(n) = 3*a(n-1) + (n-1)*a(n-2). - _Vaclav Kotesovec_, May 23 2013

%e E.g.f.: A(x) = 1 + 3*x + 10*x^2/2! + 36*x^3/3! + 138*x^4/4! + 558*x^5/5! +...

%t CoefficientList[Series[Exp[3*x + x^2/2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, May 23 2013 *)

%o (PARI) {a(n)=n!*polcoeff(exp(3*x+x^2/2+x*O(x^n)),n)}

%o (PARI) {a(n)=sum(k=0,n\2,3^(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+3*x+x*O(x^n)); for(k=1, n-1, CF=1/(1-3*x-(n-k)*x^2*CF)); polcoeff(CF, n)}

%Y Cf. A202833.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 25 2011