A377194 Expansion of 1/(1 - 4*x*(1+x)^2)^(3/2).
1, 6, 42, 266, 1650, 10032, 60202, 357744, 2109882, 12369280, 72163560, 419315340, 2428226530, 14021002860, 80757350040, 464127134636, 2662303793226, 15245389224492, 87168383093576, 497721319382220, 2838427001118456, 16168991846946656, 92012074475132892
Offset: 0
Keywords
Programs
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Mathematica
a[n_]:=Sum[(2*k+1)Binomial[2*k,k]Binomial[2k,n-k],{k,0,n}]; Array[a,23,0] (* Stefano Spezia, May 08 2025 *) -
PARI
a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(2*k, n-k));
Formula
a(0) = 1, a(1) = 6, a(2) = 42; a(n) = (2*(2*n+1)*a(n-1) + 8*(n+1)*a(n-2) + 2*(2*n+3)*a(n-3))/n.
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(2*k,n-k).