A177162 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)=6, k=0 and l=-1.

1, 6, 11, 57, 245, 1294, 6781, 37728, 213225, 1235908, 7267625, 43355213, 261455499, 1592057090, 9772992459, 60420010845, 375850271829, 2350842606832, 14775426937345, 93270580122351, 591082988357567, 3759155772624834

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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=0, l=-1).

D-finite with recurrence: (n+1)*a(n) = 2*(3*n-1)*a(n-1) - (27 - 11*n)*a(n-2) - 4*(10*n-31)*a(n-3) + 24*(n-4)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ sqrt(5*sqrt(2)-3)*(4+2*sqrt(2))^n/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

Example

a(2) = 2*1*6-1 = 11. a(3) = 2*1*11+6^2-1 = 57.

Maple

l:=-1: : k := 0 : m:=6: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

Rest[CoefficientList[Series[-Sqrt[3*x+1]*Sqrt[8*x^2-8*x+1]/(2*Sqrt[1-x]), {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 24 2012 *)

Crossrefs

Sequence in context: A271119 A271299 A177197 * A152448 A289285 A073219

Adjacent sequences: A177159 A177160 A177161 * A177163 A177164 A177165

Keyword

easy,nonn

Author

Richard Choulet, May 04 2010

Status

approved