A054441 Convolution of (shifted) A026671 with A000984 (central binomial coefficients of even order).

0, 1, 5, 23, 103, 455, 1993, 8679, 37633, 162643, 701075, 3015563, 12948083, 55513327, 237705547, 1016736115, 4344766607, 18550920063, 79149527249, 337482635279, 1438155203665, 6125448713739, 26077796587441, 110974892937943, 472081467302933, 2007534192877275, 8534465842495133

Offset: 0

Author

Wolfdieter Lang, Mar 21 2000

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A000045, A000984, A026671.

Programs

GAP

List([0..30], n-> Sum([0..n], k-> Binomial(2*n, n-k)*Fibonacci(k) )); # G. C. Greubel, Jul 15 2019

Magma

[(&+[Binomial(2*n, n-k)*Fibonacci(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jul 15 2019

Mathematica

Table[SeriesCoefficient[x/((-x+Sqrt[1-4*x])*Sqrt[1-4*x]),{x,0,n}],{n,0,30}] (* Vaclav Kotesovec, Oct 09 2012 *)

Maxima

a(n):=sum(fib(k)*binomial(2*n,n-k),k,1,n); /* Vladimir Kruchinin, Mar 19 2016 */

PARI

x='x+O('x^66); concat([0],Vec(x/((-x+sqrt(1-4*x))*sqrt(1-4*x)))) \\ Joerg Arndt, May 06 2013

Sage

[sum(binomial(2*n, n-k)*fibonacci(k) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jul 15 2019

Formula

G.f.: cbie(x)*x/(-x+1/cbie(x)), with cbie(x)=1/sqrt(1-4*x) = g.f. for A000984.

a(n) = Sum_{k=0..n} A026671(k-1)*binomial(2*(n-k), n-k), with A026671(-1):= 0.

a(n) = A026671(n) - binomial(2*n, n).

a(n) = Sum_{k=1..n} a(k-1)*binomial(2*(n-k), n-k) + 4^(n-1), n >= 1.

Recurrence: (n-2)*a(n) = 2*(4*n-9)*a(n-1) - (15*n-38)*a(n-2) - 2*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 09 2012

a(n) ~ (sqrt(5)+2)^n/sqrt(5). - Vaclav Kotesovec, Oct 09 2012

a(n) = Sum_{k=1..n} binomial(2*n,n-k)*F(k), where F denotes a Fibonacci number (A000045). - Vladimir Kruchinin, Mar 19 2016