A201366 E.g.f.: 2*exp(2*x) / (5 - 3*exp(2*x)).

1, 5, 40, 470, 7360, 144080, 3384640, 92761520, 2905461760, 102379969280, 4008411658240, 172632406008320, 8110747682652160, 412820794294292480, 22628039202542755840, 1328909797186015877120, 83247808119808161218560, 5540883903212529402183680, 390489065613179063896637440

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Formula

O.g.f.: A(x) = Sum_{n>=0} n! * 5^n*x^n / Product_{k=0..n} (1+2*k*x).

O.g.f.: A(x) = 1/(1 - 5*x/(1-3*x/(1 - 10*x/(1-6*x/(1 - 15*x/(1-9*x/(1 - 20*x/(1-12*x/(1 - 25*x/(1-15*x/(1 - ...))))))))))), a continued fraction.

a(n) = Sum_{k=0..n} (-2)^(n-k) * 5^k * Stirling2(n,k) * k!.

a(n) = Sum_{k=0..n} A123125(n,k)*5^k*3^(n-k). - Philippe Deléham, Nov 30 2011

a(n) ~ n! / (3*(log(5/3)/2)^(n+1)). - Vaclav Kotesovec, Jun 13 2013

a(n) = 2^n*log(5/3) * Integral_{x = 0..oo} (ceiling(x))^n * (5/3)^(-x) dx. - Peter Bala, Feb 06 2015

Example

E.g.f.: E(x) = 1 + 5*x + 40*x^2/2! + 470*x^3/3! + 7360*x^4/4! + 144080*x^5/5! + ...
O.g.f.: A(x) = 1 + 5*x + 40*x^2 + 470*x^3 + 7360*x^4 + 144080*x^5 + ...
where A(x) = 1 + 5*x/(1+2*x) + 2!*5^2*x^2/((1+2*x)*(1+4*x)) + 3!*5^3*x^3/((1+2*x)*(1+4*x)*(1+6*x)) + 4!*5^4*x^4/((1+2*x)*(1+4*x)*(1+6*x)*(1+8*x)) + ...

Mathematica

Table[Sum[(-2)^(n-k)*5^k*StirlingS2[n, k]*k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 13 2013 *)
With[{nn=20}, CoefficientList[Series[(2Exp[2x])/(5-3Exp[2x]), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Dec 29 2018 *)

Prog

(PARI) {a(n)=n!*polcoeff(2*exp(2*x+x*O(x^n))/(5 - 3*exp(2*x+x*O(x^n))), n)}
(PARI) {a(n)=polcoeff(sum(m=0, n, 5^m*m!*x^m/prod(k=1, m, 1+2*k*x+x*O(x^n))), n)}
(PARI) {Stirling2(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, (-2)^(n-k)*5^k*Stirling2(n, k)*k!)}

Crossrefs

Cf. A201365, A201367, A201368.

Sequence in context: A292405 A094574 A090362 * A281160 A282476 A198247

Adjacent sequences: A201363 A201364 A201365 * A201367 A201368 A201369

Keyword

nonn,easy

Author

Paul D. Hanna, Nov 30 2011

Status

approved