A385813 Expansion of 1/((1-3*x) * (1-7*x))^(3/2).
1, 15, 156, 1400, 11655, 92925, 721140, 5496300, 41361255, 308344025, 2282167272, 16795140180, 123030071437, 897791417775, 6530377362480, 47370038320800, 342794475282915, 2475479922896925, 17843821672113780, 128412824128709400, 922775179449162501, 6622378039719342615
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
Programs
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Magma
R
:= PowerSeriesRing(Rationals(), 34); f := 1/((1-3*x) * (1-7*x))^(3/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 25 2025 -
Mathematica
CoefficientList[Series[1/((1-3x)*(1-7*x))^(3/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 25 2025 *) -
PARI
my(N=30, x='x+O('x^N)); Vec(1/((1-3*x)*(1-7*x))^(3/2))
Formula
n*a(n) = (10*n+5)*a(n-1) - 21*(n+1)*a(n-2) for n > 1.
a(n) = (1/4)^n * Sum_{k=0..n} 3^k * 7^(n-k) * (2*k+1) * (2*(n-k)+1) * binomial(2*k,k) * binomial(2*(n-k),n-k).
a(n) = Sum_{k=0..n} 3^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = Sum_{k=0..n} (-1)^k * 7^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(n+2,n-k).
a(n) = binomial(n+2,2) * A182401(n).
a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = Sum_{k=0..n} (5/2)^k * (-21/10)^(n-k) * (2*k+1) * binomial(2*k,k) * binomial(k,n-k).