A374506 Expansion of 1/(1 - 2*x - 3*x^2)^(7/2).
1, 7, 42, 210, 966, 4158, 17094, 67782, 261261, 983983, 3635632, 13217568, 47393892, 167919948, 588772152, 2045481480, 7048466271, 24111291897, 81939285582, 276810647190, 930096277110, 3109797881190, 10350813392010, 34309326304890, 113288127469335
Offset: 0
Programs
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Mathematica
a[n_]:= Pochhammer[n+1, 6]*Hypergeometric2F1[(1-n)/2, -n/2, 4, 4]/6!; Array[a,25,0] (* Stefano Spezia, Jul 10 2024 *)
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PARI
a(n) = binomial(n+6, 3)/20*sum(k=0, n\2, binomial(n+3, n-2*k)*binomial(2*k+3, k));
Formula
a(0) = 1, a(1) = 7; a(n) = ((2*n+5)*a(n-1) + 3*(n+5)*a(n-2))/n.
a(n) = (binomial(n+6,3)/20) * Sum_{k=0..floor(n/2)} binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = Pochhammer(n+1, 6)*hypergeom([(1-n)/2, -n/2], [4], 4)/6!. - Stefano Spezia, Jul 10 2024
a(n) = Sum_{k=0..n} (-2)^k * (3/2)^(n-k) * binomial(-7/2,k) * binomial(k,n-k). - Seiichi Manyama, Aug 23 2025