A108490 Expansion of 1/sqrt(1-4x-8x^2-24x^3+36x^4).

1, 2, 10, 56, 268, 1448, 7864, 42752, 236368, 1313696, 7339552, 41217920, 232321984, 1313731712, 7449834880, 42347380736, 241225384192, 1376662561280, 7869527190016, 45051709749248, 258256281078784, 1482218578159616

Offset: 0

Author

Paul Barry, Jun 04 2005

Keywords

• easy
• nonn

Comments

In general, Sum_{k=0..n} C(n-k,k)^2*a^k*b^(n-k) has expansion 1/sqrt(1-2bx-(2ab-b^2)x^2-2a*b^2*x^3+(ab)^2*x^4).

Links

• Robert Israel, Table of n, a(n) for n = 0..1301
• Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Crossrefs

Cf. A108486.

Programs

Maple

f:= gfun:-rectoproc({(72 + 36*n)*a(n) + (-60 - 24*n)*a(n + 1) + (-8*n - 24)*a(n + 2) + (-14 - 4*n)*a(n + 3) + (n + 4)*a(n + 4), a(0) = 1, a(1) = 2, a(2) = 10, a(3) = 56}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Jan 07 2019

Mathematica

CoefficientList[Series[1/Sqrt[1-4x-8x^2-24x^3+36x^4], {x,0,30}], x]  (* Harvey P. Dale, Mar 13 2011 *)

Formula

a(n) = Sum_{k=0..n} C(n-k, k)^2*3^k*2^(n-k).

D-finite with recurrence: n*a(n) +2*(-2*n+1)*a(n-1) +8*(-n+1)*a(n-2) +12*(-2*n+3)*a(n-3) +36*(n-2)*a(n-4)=0. - R. J. Mathar, Feb 20 2015

Recurrence confirmed using d.e. (72*x^3 - 36*x^2 - 8*x - 2)*g(x) + (36*x^4 - 24*x^3 - 8*x^2 - 4*x + 1)*g'(x) = 0 satisfied by the g.f.. - Robert Israel, Jan 07 2019