A374488 Expansion of 1/(1 - 2*x - 11*x^2)^(3/2).
1, 3, 24, 100, 555, 2541, 12628, 59004, 281655, 1315765, 6171132, 28692456, 133315273, 616780815, 2848833120, 13124483344, 60364983987, 277142478921, 1270586298520, 5817063737100, 26600252408961, 121501917998263, 554429553154044, 2527595449990500
Offset: 0
Links
- Paolo Xausa, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
Module[{x}, CoefficientList[Series[1/(1 - (11*x + 2)*x)^(3/2), {x, 0, 25}], x]] (* Paolo Xausa, Aug 25 2025 *) -
PARI
a(n) = binomial(n+2, 2)*sum(k=0, n\2, 3^k*binomial(n, 2*k)*binomial(2*k, k)/(k+1));
Formula
a(0) = 1, a(1) = 3; a(n) = ((2*n+1)*a(n-1) + 11*(n+1)*a(n-2))/n.
a(n) = binomial(n+2,2) * A025237(n).
From Seiichi Manyama, Aug 20 2025: (Start)
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 3^k * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (1/2)^k * (11/2)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k). (End)
a(n) ~ sqrt(n) * (1 + 2*sqrt(3))^(n + 3/2) / (4 * 3^(3/4) * sqrt(Pi)). - Vaclav Kotesovec, Aug 21 2025