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

0,3

Links

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

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

Mathematica

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

Prog

(PARI) x='x+O('x^66); concat([0], Vec(x/((-x+sqrt(1-4*x))*sqrt(1-4*x)))) \\ Joerg Arndt, May 06 2013
(Maxima)
a(n):=sum(fib(k)*binomial(2*n, n-k), k, 1, n); /* Vladimir Kruchinin, Mar 19 2016 */
(Magma) [(&+[Binomial(2*n, n-k)*Fibonacci(k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jul 15 2019
(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
(GAP) List([0..30], n-> Sum([0..n], k-> Binomial(2*n, n-k)*Fibonacci(k) )); # G. C. Greubel, Jul 15 2019

Crossrefs

Cf. A000045, A000984, A026671.

Sequence in context: A258431 A356340 A120902 * A289803 A102285 A218985

Adjacent sequences: A054438 A054439 A054440 * A054442 A054443 A054444

Keyword

nonn,easy

Author

Wolfdieter Lang, Mar 21 2000

Status

approved