A377198 Expansion of 1/(1 - 4*x/(1-x)^2)^(3/2).
1, 6, 42, 278, 1794, 11382, 71338, 443046, 2732034, 16751462, 102235050, 621535158, 3766261506, 22758222294, 137186860842, 825211984710, 4954574749698, 29697908825286, 177746214414634, 1062416305340502, 6342559258130178, 37823152988963126, 225328426205608362
Offset: 0
Keywords
Links
- Stefano Spezia, Table of n, a(n) for n = 0..1200
Programs
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Magma
R
:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x)^2)^(3/2))); // Vincenzo Librandi, May 11 2025 -
Mathematica
Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n+k-1,n-k],{k,0,n}],{n,0,35}] (* Vincenzo Librandi, May 11 2025 *) -
PARI
a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(n+k-1, n-k));
Formula
a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (3-k/n) * (n-k) * a(k).
a(n) = ((7*n-1)*a(n-1) - (7*n-20)*a(n-2) + (n-3)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+k-1,n-k).
a(n) ~ 2^(1/4) * sqrt(n) * (1 + sqrt(2))^(2*n) / sqrt(Pi). - Vaclav Kotesovec, May 03 2025
a(n) = 6*n*hypergeom([5/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, May 08 2025