|
%I #35 Sep 08 2022 08:45:01
%S 0,1,5,23,103,455,1993,8679,37633,162643,701075,3015563,12948083,
%T 55513327,237705547,1016736115,4344766607,18550920063,79149527249,
%U 337482635279,1438155203665,6125448713739,26077796587441,110974892937943,472081467302933,2007534192877275,8534465842495133
%N Convolution of (shifted) A026671 with A000984 (central binomial coefficients of even order).
%H Vincenzo Librandi, <a href="/A054441/b054441.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: cbie(x)*x/(-x+1/cbie(x)), with cbie(x)=1/sqrt(1-4*x) = g.f. for A000984.
%F a(n) = Sum_{k=0..n} A026671(k-1)*binomial(2*(n-k), n-k), with A026671(-1):= 0.
%F a(n) = A026671(n) - binomial(2*n, n).
%F a(n) = Sum_{k=1..n} a(k-1)*binomial(2*(n-k), n-k) + 4^(n-1), n >= 1.
%F 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
%F a(n) ~ (sqrt(5)+2)^n/sqrt(5). - _Vaclav Kotesovec_, Oct 09 2012
%F 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
%t Table[SeriesCoefficient[x/((-x+Sqrt[1-4*x])*Sqrt[1-4*x]),{x,0,n}],{n,0,30}] (* _Vaclav Kotesovec_, Oct 09 2012 *)
%o (PARI) x='x+O('x^66); concat([0],Vec(x/((-x+sqrt(1-4*x))*sqrt(1-4*x)))) \\ _Joerg Arndt_, May 06 2013
%o (Maxima)
%o a(n):=sum(fib(k)*binomial(2*n,n-k),k,1,n); /* _Vladimir Kruchinin_, Mar 19 2016 */
%o (Magma) [(&+[Binomial(2*n, n-k)*Fibonacci(k): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jul 15 2019
%o (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
%o (GAP) List([0..30], n-> Sum([0..n], k-> Binomial(2*n, n-k)*Fibonacci(k) )); # _G. C. Greubel_, Jul 15 2019
%Y Cf. A000045, A000984, A026671.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Mar 21 2000
|
