A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

1, 4, 40, 480, 6240, 84864, 1188096, 16972800, 246105600, 3609548800, 53421322240, 796463349760, 11946950246400, 180123249868800, 2727580640870400, 41459225741230080, 632253192553758720

Offset: 0

Author

Wolfdieter Lang

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• A. Straub, V. H. Moll, and T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009) 31-41, eq (1.10).

Crossrefs

Cf. A007696.

Expansion of (1-b^2*x)^(-1/b): A000984 (b=2), A004987 (b=3), this sequence (b=4), A034688 (b=5), A004993 (b=6), A034835 (b=7), A034977 (b=8), A035024 (b=9), A035308 (b=10).

Programs

Mathematica

CoefficientList[Series[1/Surd[1-16x,4],{x,0,20}],x] (* Harvey P. Dale, Aug 06 2018 *)

Formula

a(n) = (4^n/n!)*A007696(n), n >= 1, a(0) := 1, A007696(n) = (4*n-3)!^4 := Product_{j = 1..n} 4*j - 3.

G.f.: (1 - 16*x)^(-1/4).

D-finite with recurrence: n*a(n) + 4*(-4*n + 3)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020

From Peter Bala, Mar 31 2024: (Start)

a(n) = (-16)^n*binomial(-1/4, n).

a(n) ~ Gamma(3/4)/(sqrt(2)*Pi) * 16^n/n^(3/4).

E.g.f.: hypergeom([1/4], [1], 16*x).

a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-1/4, k)* binomial(-1/4, 2*n - k).

(16^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).

Sum_{k = 0..n} a(k)*a(n-k) = (4^n)*binomial(2*n, n) = A098430.

Sum_{k = 0..2*n} a(k)*a(2*n-k) = (16^n)*binomial(4*n, 2*n). (End)