A387343 Expansion of 1/(1 - 8*x + 4*x^2)^(5/2).

1, 20, 270, 3080, 31990, 312984, 2937900, 26751120, 237977190, 2078447800, 17884238372, 152002796400, 1278603975740, 10660760170480, 88213513627800, 725107271106336, 5925674432448390, 48175954959638520, 389871795632108020, 3142078444590396080, 25228464363569709396

Offset: 0

Author

Seiichi Manyama, Aug 27 2025

Keywords

• nonn
• new

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..800

Crossrefs

Cf. A069835, A331515, A387344.

Cf. A002802, A387313, A387341.

Cf. A387339.

Programs

Magma

R := PowerSeriesRing(Rationals(), 34); f := 1/(1 - 8*x + 4*x^2)^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025

Mathematica

CoefficientList[Series[1/(1-8*x+4*x^2)^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(1/(1-8*x+4*x^2)^(5/2))

Formula

n*a(n) = 4*(2*n+3)*a(n-1) - 4*(n+3)*a(n-2) for n > 1.

a(n) = (binomial(n+4,2)/6) * A387339(n).

a(n) = (-1)^n * Sum_{k=0..n} (1/2)^(n-4*k) * binomial(-5/2,k) * binomial(k,n-k).