A382540 Expansion of 1/(1 - x/(1 - 4*x)^(1/2))^3.
1, 3, 12, 52, 231, 1035, 4650, 20898, 93849, 420935, 1885248, 8430588, 37642819, 167824905, 747143298, 3321632498, 14747814597, 65397373761, 289652172896, 1281454446408, 5663228541975, 25002457308487, 110275917725658, 485935158536874, 2139412626785505
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..600
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1 - 4*x)^(1/2))^3)); // Vincenzo Librandi, May 12 2025 -
Mathematica
Table[Sum[(4)^(n-k)* Binomial[k+2,2]*Binomial[n-k/2-1, n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 12 2025 *) -
PARI
a(n) = sum(k=0, n, 4^(n-k)*binomial(k+2, 2)*binomial(n-k/2-1, n-k));
Formula
a(n) = Sum_{k=0..n} 4^(n-k) * binomial(k+2,2) * binomial(n-k/2-1,n-k).
D-finite with recurrence 3*(-n+1)*a(n) +6*(6*n-11)*a(n-1) +2*(-71*n+199)*a(n-2) +4*(44*n-183)*a(n-3) +3*(11*n+5)*a(n-4) +2*(-2*n+7)*a(n-5)=0. - R. J. Mathar, Apr 02 2025