A387316 Expansion of 1/((1-x) * (1-13*x))^(7/2).
1, 49, 1498, 36750, 792246, 15681666, 292137846, 5201141946, 89399571261, 1494080348761, 24403114463728, 391038174645664, 6165638429715492, 95880046644705876, 1473241291627666488, 22401020288076984120, 337479336374849120991, 5042656883996693680719
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))^(7/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025 -
Mathematica
CoefficientList[Series[1/((1-x)*(1-13*x))^(7/2),{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))^(7/2))
Formula
n*a(n) = (14*n+35)*a(n-1) - 13*(n+5)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 13^k * binomial(-7/2,k) * binomial(-7/2,n-k).
a(n) = Sum_{k=0..n} (-12)^k * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = Sum_{k=0..n} 12^k * 13^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = (binomial(n+6,3)/20) * A387311(n).
a(n) = (-1)^n * Sum_{k=0..n} 14^k * (13/14)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).