A387313 Expansion of 1/((1-x) * (1-9*x))^(5/2).
1, 25, 415, 5775, 72870, 864150, 9818130, 108109650, 1162302735, 12262882775, 127424209913, 1307536637225, 13276264807260, 133597932407100, 1334029357684980, 13231465264538100, 130461712570627245, 1279632533997010725, 12492837802976030115, 121456026730456739475
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..800
Programs
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Magma
R
:= PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-9*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025 -
Mathematica
CoefficientList[Series[1/((1-x)*(1-9*x))^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-9*x))^(5/2))
Formula
n*a(n) = (10*n+15)*a(n-1) - 9*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 9^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-8)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 8^k * 9^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * A387307(n).
a(n) = (-1)^n * Sum_{k=0..n} 10^k * (9/10)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).