A026908 a(n) = A026907(2*n, n).

1, 44, 198, 758, 2824, 10530, 39516, 149274, 567120, 2164898, 8297206, 31907214, 123053938, 475753142, 1843357752, 7155936162, 27826486272, 108369397602, 422614721742, 1650118887102, 6450143419962, 25238550057222

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• changed

Links

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

Crossrefs

Cf. A000108, A026907.

Programs

Magma

A026908:= func< n | (n+1)*Catalan(n) +3*(n+3)*Catalan(n+2) -18 >;
[A026908(n): n in [0..40]]; // G. C. Greubel, Aug 22 2025

Mathematica

With[{b=Binomial}, Table[b[2*n,n] +3*b[2*n+4,n+2] -18, {n,0,40}]] (* G. C. Greubel, Aug 22 2025 *)

SageMath

def A026908(n): return binomial(2*n,n) +3*binomial(2*n+4,n+2) -18
print([A026908(n) for n in range(41)]) # G. C. Greubel, Aug 22 2025

Formula

a(n) = 4^n*Gamma(n+1/2)*(38 + 99*n + 49*n^2)*Pi^(-1/2)/Gamma(n+3) - 18 (based on guessed recurrence) - Mark van Hoeij, Oct 30 2011

From G. C. Greubel, Aug 22 2025: (Start)

a(n) = binomial(2*n, n) + 3*binomial(2*n+4, n+2) - 18.

a(n) = (n+2)*A000108(n) + 3*(n+3)*A000108(n+2) - 18.

G.f.: 2*(7 + 45*x + 2*x^2 - (5 - 23*x)*sqrt(1-4*x))/((1-x)*sqrt(1-4*x)*(1 + sqrt(1 - 4*x))^2).

E.g.f.: (1/x)*exp(2*x)*( 25*x*BesselI(0, 2*x) - 6*(1-4*x)*BesselI(1, 2*x) ) - 18*exp(x). (End)