A110313 Expansion of e.g.f. exp(x)/(1-x-x^2).

1, 2, 7, 34, 221, 1786, 17347, 196442, 2542969, 37030546, 599172671, 10664259442, 207061905877, 4355429249354, 98661276360571, 2394559287772906, 61991654930903537, 1705178260099590562, 49662655090649112439, 1526761411676393108546, 49407037167974524897741

Offset: 0

Author

Paul Barry, Jul 19 2005

Keywords

• easy
• nonn

Comments

Binomial transform of A005442.

Rows sums of A039948.

Links

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

Crossrefs

Cf. A000045, A005442, A039948 (row sums).

Programs

Magma

[(&+[Binomial(n,k)*Factorial(k)*Fibonacci(k+1): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Nov 30 2021

Mathematica

Table[Sum[Binomial[n, k]k!Fibonacci[k+1], {k,0,n}], {n,0,30}] (* Emanuele Munarini, May 27 2011 *)
With[{nn=20},CoefficientList[Series[Exp[x]/(1-x-x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 09 2023 *)

Maxima

makelist(sum(binomial(n,k)*k!*fib(k+1), k,0,n), n,0,30); /* Emanuele Munarini, May 27 2011 */

Sage

[sum(factorial(n)*fibonacci(n-k+1)/factorial(k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Nov 30 2021

Formula

E.g.f.: exp(x)/(1-x-x^2).

a(n) = Sum_{k=0..n} binomial(n,k)*k!*Fibonacci(k+1).

a(n) = Sum_{k=0..n} n!*Fibonacci(n-k+1)/k!.

Recurrence: a(n+3) = (n+4)*a(n+2) + (n+2)^2*a(n+1) - (n+2)*(n+1)*a(n). - Emanuele Munarini, May 27 2011

a(n) ~ n!*exp((sqrt(5)-1)/2)*((1+sqrt(5))/2)^(n+1)/sqrt(5). - Vaclav Kotesovec, Oct 18 2012