A177110 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=0, k=-2 and l=0.

1, 0, -4, -14, -36, -66, -32, 406, 2332, 8046, 18472, 13558, -127580, -848722, -3236272, -8208026, -8089908, 51660014, 389206456, 1588065494, 4318743220, 5225603310, -23415860512, -200117753530, -863731324836, -2486101104594, -3511206832184, 11171231626806, 110034261679044, 500062779185198

Offset

0,3

Links

Robert Israel, Table of n, a(n) for n = 0..1739

Formula

G.f: (1-z + sqrt(1-6*z+13*z^2+4*z^3-4*z^4))/(2*(z-z^2)).

D-finite with recurrence: +(n+1)*a(n) +(-7*n+2)*a(n-1) +(19*n-29)*a(n-2) +3*(-3*n+4)*a(n-3) +2*(-4*n+17)*a(n-4) +4*(n-5)*a(n-5)=0. - R. J. Mathar, Mar 02 2016

Recurrence follows from the differential equation -1+5*z+3*z^2-5*z^3+2*z^4 + (1-5*z+9*z^2-15*z^3+2*z^4)*g(z) + (z-7*z^2+19*z^3-9*z^4-8*z^5+4*z^6)*g'(z) satisfied by the g.f. - Robert Israel, Jul 14 2017

Example

a(2)=2*1*0-4=-4. a(3)=2*1*(-4)-4+0^2-2=-14. a(4)=2*1*(-14)-4+2*0*(-4)-4=-36.

Maple

l:=0: : k := -2 : m:=0:d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 30); seq(d(n), n=0..30);

Crossrefs

Sequence in context: A288678 A295180 A305906 * A213045 A061989 A079908

Adjacent sequences: A177107 A177108 A177109 * A177111 A177112 A177113

Keyword

easy,sign

Author

Richard Choulet, May 03 2010

Extensions

G.f. edited, and more terms, from Robert Israel, Jul 14 2017

Status

approved