A110313 - OEIS

%I #28 Sep 22 2025 16:00:47

%S 1,2,7,34,221,1786,17347,196442,2542969,37030546,599172671,

%T 10664259442,207061905877,4355429249354,98661276360571,

%U 2394559287772906,61991654930903537,1705178260099590562,49662655090649112439,1526761411676393108546,49407037167974524897741

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

%C Binomial transform of A005442.

%C Rows sums of A039948.

%H Vincenzo Librandi, <a href="/A110313/b110313.txt">Table of n, a(n) for n = 0..200</a>

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

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

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

%F 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

%F a(n) ~ n!*exp((sqrt(5)-1)/2)*((1+sqrt(5))/2)^(n+1)/sqrt(5). - _Vaclav Kotesovec_, Oct 18 2012

%t Table[Sum[Binomial[n, k]k!Fibonacci[k+1], {k,0,n}], {n,0,30}] (* _Emanuele Munarini_, May 27 2011 *)

%t With[{nn=20},CoefficientList[Series[Exp[x]/(1-x-x^2),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 09 2023 *)

%o (Maxima) makelist(sum(binomial(n,k)*k!*fib(k+1), k,0,n), n,0,30); /* _Emanuele Munarini_, May 27 2011 */

%o (Magma) [(&+[Binomial(n,k)*Factorial(k)*Fibonacci(k+1): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Nov 30 2021

%o (SageMath) [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

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

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jul 19 2005