A377186 Expansion of 1/(1 - 4*x^2 - 4*x^3)^(3/2).
1, 0, 6, 6, 30, 60, 170, 420, 1050, 2660, 6552, 16380, 40362, 99792, 245520, 603372, 1480050, 3624192, 8863712, 21647340, 52811616, 128700000, 313341756, 762206016, 1852565650, 4499346072, 10919990460, 26485897932, 64201490352, 155536089240, 376606931436
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1500
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^2 - 4*x^3)^(3/2))); // Vincenzo Librandi, May 11 2025 -
Mathematica
Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[k,n-2*k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, May 11 2025 *) -
PARI
a(n) = sum(k=0, n\2, (2*k+1)*binomial(2*k, k)*binomial(k, n-2*k));
Formula
a(0) = 1, a(1) = 0, a(2) = 6; a(n) = (4*(n+1)*a(n-2) + 2*(2*n+3)*a(n-3))/n.
a(n) = Sum_{k=0..floor(n/2)} (2*k+1) * binomial(2*k,k) * binomial(k,n-2*k).