A176645 a(n+1) = m + Sum_{j=0..n} (a(j)*a(n-j) + k) for n>=1, with a(0)=1, a(1)=4, k=1 and m=1.

1, 4, 11, 42, 177, 817, 3981, 20164, 105027, 558915, 3025287, 16603039, 92169795, 516644985, 2920055107, 16622691454, 95220681081, 548477688005, 3174801937437, 18457766735525, 107734640321681, 631075890235811

Offset: 0

Author

Richard Choulet, Apr 22 2010

Keywords

• easy
• nonn

Examples

			a(2) = 2*1*4 + 2 + 1 = 11.
a(3) = 2*1*11 + 2 + 4^2 + 1 + 1 = 42.
a(4) = 2*1*42 + 2 + 2*4*11 + 2 + 1 = 177.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A176612.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40);
Coefficients(R!( (1-x -Sqrt(1 -6*x -3*x^2 +8*x^3 -4*x^4))/(2*x*(1-x)) )); // G. C. Greubel, Jul 02 2021

Maple

# First program
l:=1: : k := 1 : m:=4: a(0):=1 : a(1):=m:
for n from 1 to 51 do a(n+1):=sum(a(p)*a(n-p)+k, p=0..n) +l : od :
seq(a(n), n=0..40);
# Second program
n:= 40;
S:= series((1-x -sqrt(-4*x^4 +8*x^3 -3*x^2 -6*x +1))/(2*x*(1-x)), x, n+1);
seq(coeff(S, x, j), j = 0..n); # modified by G. C. Greubel, Jul 02 2021

Mathematica

a[n_,k_,m_]:= a[n,k,m]= If[n<2, 4^n, m +k*n +Sum[a[j,k,m]*a[n-j-1,k,m], {j,0,n-1}]];
Table[a[n, 1, 1], {n, 0, 40}] (* G. C. Greubel, Jul 02 2021 *)

Sage

@CachedFunction
def a(n,k,m): return 4^n if (n<2) else m + k*n + sum(a(j,k,m)*a(n-j-1,k,m) for j in (0..n-1))
[a(n,1,1) for n in (0..40)] # G. C. Greubel, Jul 02 2021

Formula

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

(n+1)*a(n) - (7*n-2)*a(n-1) + 3*(n+1)*a(n-2) + (11*n-34)*a(n-3) - 12*(n-4)*a(n-4) + 4*(n-5)*a(n-5) = 0. - R. J. Mathar, Mar 01 2016

From G. C. Greubel, Jul 02 2021: (Start)

a(n) = m + k*n + Sum_{j=0..n-1} a(j)*a(n-j-1), with a(0)=1, a(1)=4, k=1 and m=1.

G.f.: (1-x -sqrt(1 -6*x -3*x^2 +8*x^3 -4*x^4))/(2*x*(1-x)). (End)