A387315 Expansion of 1/((1-x) * (1-13*x))^(5/2).
1, 35, 825, 16415, 297220, 5067972, 82893720, 1315073760, 20381376015, 310101196405, 4648184007467, 68817616687365, 1008344472704660, 14644604899082620, 211073938188085620, 3022082811670829676, 43017189132931007655, 609159438493806780405, 8586490781973282553375
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) * (1-13*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025 -
Mathematica
CoefficientList[Series[1/((1-x)*(1-13*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)*(1-13*x))^(5/2))
Formula
n*a(n) = (14*n+21)*a(n-1) - 13*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 13^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-12)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 12^k * 13^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * A387310(n).
a(n) = (-1)^n * Sum_{k=0..n} 14^k * (13/14)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).