A073701 a(n) = n^2*a(n-1)+(-1)^n.

1, 0, 1, 8, 129, 3224, 116065, 5687184, 363979777, 29482361936, 2948236193601, 356736579425720, 51370067437303681, 8681541396904322088, 1701582113793247129249, 382855975603480604081024, 98011129754491034644742145, 28325216499047909012330479904, 9177370145691522519995075488897

Offset: 0

Author

Vladeta Jovovic, Aug 30 2002

Keywords

• nonn

Comments

The sequence b(n) := n!^2 satisfies the same recurrence below for a(n) with the initial conditions b(0) = 1, b(1) = 1. It follows that, for n >=3, a(n) = n!^2*(1/(4 + 4/(8 + 9/(15 +...+ (n-1)^2/(n^2-1))))). Hence BesselJ(0,2) := sum {k = 0..inf} (-1)^k/k!^2 = 1/(4 + 4/(8 + 9/(15 + ...+(n-1)^2/(n^2+1 + ...)))) = 0.22388 90779 ... . Cf. A006040. - Peter Bala, Jul 09 2008

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..253

Crossrefs

Cf. A000166, A006040.

Programs

Magma

[1] cat [ n eq 1 select 0 else n^2*Self(n-1)+(-1)^n:n in [1..15]]; // Marius A. Burtea, Feb 13 2020

Mathematica

Join[{a = 1}, Table[a = a*n^2 + (-1)^n, {n, 15}]] (* Jayanta Basu, Jul 08 2013 *)

Formula

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

BesselJ(0, 2*sqrt(x))/(1-x) = Sum_{n>=0} a(n)*x^n/n!^2.

a(n) = round(n!^2*BesselJ(0, 2)), n>0.

Recurrence: a(0) = 1, a(1) = 0, a(n) = (n^2-1)*a(n-1) + (n-1)^2*a(n-2), n >= 2. - Peter Bala, Jul 09 2008, corrected by Georg Fischer, Feb 13 2020

a(n) = Sum_{k=0..n} (-1)^(n-k)*(k!*binomial(n,k))^2. - Ridouane Oudra, Jul 11 2025