A382542 Expansion of 1/(1 - x/(1 - 4*x)^(3/2))^3.
1, 3, 24, 172, 1191, 8091, 54214, 359274, 2358945, 15365815, 99399132, 639081780, 4086689187, 26006041209, 164767882902, 1039787209898, 6537976304109, 40973438195025, 255998969164612, 1594973077037136, 9911483124031335, 61443351455986359, 380044418794190118
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)^(3/2))^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,0,25}] (* 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 (1951*n-648)*(n-1)*a(n) +2*(-19507*n^2+55239*n-42212)*a(n-1) +(310113*n^2-1357025*n+1557754)*a(n-2) +2*(-616231*n^2+3607803*n-5547102)*a(n-3) +4*(616138*n^2-4491203*n+8306122)*a(n-4) -256*(3899*n-16903)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Apr 02 2025