A387281 - OEIS

%I #16 Aug 25 2025 09:25:01

%S 1,28,462,5880,63966,626472,5692764,48919728,402648246,3202791592,

%T 24780247492,187393703952,1390208264172,10146829592592,73029572999352,

%U 519260074512480,3652939914500646,25457292175929768,175932472247239092,1206772898939860560,8221969006750158660

%N Expansion of 1/((1-2*x) * (1-6*x))^(7/2).

%H Paolo Xausa, <a href="/A387281/b387281.txt">Table of n, a(n) for n = 0..1000</a>

%F n*a(n) = (8*n+20)*a(n-1) - 12*(n+5)*a(n-2) for n > 1.

%F a(n) = (-2)^n * Sum_{k=0..n} 3^k * binomial(-7/2,k) * binomial(-7/2,n-k).

%F a(n) = 2^n * Sum_{k=0..n} (-2)^k * binomial(-7/2,k) * binomial(n+6,n-k).

%F a(n) = Sum_{k=0..n} 4^k * 6^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).

%F a(n) = (binomial(n+6,3)/20) * A387273(n).

%F a(n) = (-1)^n * Sum_{k=0..n} 8^k * (3/2)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).

%t Module[{x}, CoefficientList[Series[1/((3*x - 2)*4*x + 1)^(7/2), {x, 0, 25}], x]] (* _Paolo Xausa_, Aug 25 2025 *)

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/((1-2*x)*(1-6*x))^(7/2))

%Y Cf. A387273.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Aug 24 2025