A112243 Expansion of exp(x*(1+x)/(1-2*x)).

1, 1, 7, 55, 577, 7441, 113671, 2003527, 39971905, 889608097, 21834577351, 585555975511, 17027451783937, 533460597334705, 17908302027585607, 641152804988733031, 24380543011087797121, 981149507717921468737, 41653436572936172408455, 1860174362332664149119607

Offset: 0

Author

Paul Barry, Aug 29 2005

Keywords

• easy
• nonn

Comments

In general, e.g.f. exp(x(1+ax)/(1-bx)) has general term sum{i=0..n, sum{j=0..n, a^j*b^(n-i-j)*C(i,j)C(n-j-1,n-i-j)*n!/i!}}.

Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is a purely periodic sequence whose period divides k. For example, taken modulo 14 the sequence becomes [1, 1, 7, 13, 3, 7, 5, 1, 1, 7, 13, 3, 7, 5, ...], a periodic sequence of period 7. Cf. A047974. - Peter Bala, Jul 15 2022

Programs

Mathematica

With[{nn=20},CoefficientList[Series[Exp[(x(x+1))/(1-2x)],{x,0,nn}], x] Range[0,nn]!] (* Harvey P. Dale, Sep 21 2011 *)

Formula

E.g.f.: exp(x*(1+x)/(1-2*x)).

a(n) = Sum_{i = 0..n} Sum_{j = 0..n} 2^(n-i-j)*C(i, j)*C(n-j-1, n-i-j)*n!/i!.

a(n) ~ 3^(1/4) * 2^(n-1) * n^(n-1/4) * exp(sqrt(3*n)-n-5/8). - Vaclav Kotesovec, Sep 25 2013

Conjecture: a(n) +(-4*n+3)*a(n-1) +2*(n-1)*(2*n-5)*a(n-2) +2*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Feb 20 2015

The e.g.f. A(x) satisfies the differential equation (4*x^2 - 4*x + 1)*A'(x) + (2*x^2 - 2*x - 1)*A(x) = 0 with A(0) = 1. Mathar's conjectured recurrence follows easily from this. - Peter Bala, Jul 15 2022