A202834 E.g.f.: exp(3*x + x^2/2).

1, 3, 10, 36, 138, 558, 2364, 10440, 47868, 227124, 1112184, 5607792, 29057400, 154465704, 841143312, 4685949792, 26674999056, 155000193840, 918475565472, 5545430185536, 34087326300576, 213170582612448, 1355345600149440, 8755789617922176, 57440317657203648

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Yasushi Ieno, A newly-generalized problem from a problem for the Mathematical Olympiad and the methods to solve it, arXiv:2012.12655 [math.GM], 2020. See p. 9.

Formula

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

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.

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

Recurrence: a(n) = 3*a(n-1) + (n-1)*a(n-2). - Vaclav Kotesovec, May 23 2013

Example

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

Mathematica

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

Prog

(PARI) {a(n)=n!*polcoeff(exp(3*x+x^2/2+x*O(x^n)), n)}
(PARI) {a(n)=sum(k=0, n\2, 3^(n-2*k)/2^k * n!/((n-2*k)!*k!))}
(PARI) /* O.g.f. as a continued fraction: */
{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)}

Crossrefs

Cf. A202833.

Sequence in context: A340941 A149041 A307346 * A129247 A162162 A149042

Adjacent sequences: A202831 A202832 A202833 * A202835 A202836 A202837

Keyword

nonn

Author

Paul D. Hanna, Dec 25 2011

Status

approved