A177180 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)=10, k=1 and l=-1.

1, 10, 21, 144, 711, 4747, 29767, 205078, 1409645, 10043729, 72216773, 528438373, 3903255409, 29138576719, 219209569841, 1661343858524, 12668020020047, 97135000445375, 748428139988567, 5792032911677831, 45000447097568843

Offset: 0

Author

Richard Choulet, May 04 2010

Keywords

• easy
• nonn

Crossrefs

Cf. A177179.

Programs

Maple

l:=-1: : k := 1 : for m from 0 to 10 do 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): od;

Formula

G.f f: f(z)=(1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=1, l=-1).

Conjecture: (n+1)*a(n) +(2-7*n)*a(n-1) +3*(17-7*n)*a(n-2) +(91*n-278)*a(n-3) +4*(101-25*n)*a(n-4) +36*(n-5)*a(n-5)=0. - R. J. Mathar, Jul 24 2012