A176830 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)=4, k=1 and l=-1.

1, 4, 9, 36, 147, 667, 3163, 15610, 79121, 409697, 2157185, 11514865, 62167261, 338870851, 1862411885, 10308811304, 57417591755, 321563019095, 1809699812171, 10229230730987, 58047787492463, 330576576766193

Offset

0,2

Links

Table of n, a(n) for n=0..21.

Formula

G.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) +(-7*n+2)*a(n-1) +3*(n+1)*a(n-2) +(19*n-62)*a(n-3) +4*(-7*n+29)*a(n-4) +12*(n-5)*a(n-5)=0. - R. J. Mathar, Feb 18 2016

Example

a(2)=2*1*4+2-1=9. a(3)=2*1*9+2+4^2+1-1=36. a(4)=2*1*36+2+2*4*9+2-1=147.

Maple

l:=-1: : k := 1 : m:=4: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);

Mathematica

CoefficientList[Series[(1+Sqrt[1-x (6+x (3+4 x (-4+3 x)))]/(-1+x))/ (2 x), {x, 0, 30}], x] (* Harvey P. Dale, May 11 2011 *)

Crossrefs

Cf. A176829.

Sequence in context: A227253 A029997 A118548 * A077211 A073977 A055872

Adjacent sequences: A176827 A176828 A176829 * A176831 A176832 A176833

Keyword

easy,nonn

Author

Richard Choulet, Apr 27 2010

Status

approved