A026909 a(n) = (1/2)*A026907(2*n, n).

22, 99, 379, 1412, 5265, 19758, 74637, 283560, 1082449, 4148603, 15953607, 61526969, 237876571, 921678876, 3577968081, 13913243136, 54184698801, 211307360871, 825059443551, 3225071709981, 12619275028611, 49423455006501, 193732625020419, 760001601263697, 2983614441691035, 11720995167614703, 46074369462135607

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• changed

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000108, A026907, A026908.

Programs

Magma

A026909:= func< n | ((n+1)*Catalan(n) +3*(n+3)*Catalan(n+2))/2 -9 >;
[A026909(n): n in [1..40]]; // G. C. Greubel, Aug 22 2025

Mathematica

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

SageMath

def A026909(n): return (binomial(2*n,n) +3*binomial(2*n+4,n+2))//2 -9
print([A026909(n) for n in range(1,41)]) # G. C. Greubel, Aug 22 2025

Formula

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

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

a(n) = ((n+2)*A000108(n) + 3*(n+3)*A000108(n+2))/2 - 9.

G.f.: 2*(3 + 25*x - x^2 - (3 - 13*x + x^2)*sqrt(1-4*x))/((1-x)*sqrt(1-4*x)*(1 + sqrt(1 - 4*x))^2).

E.g.f.: (1/(2*x))*exp(2*x)*( 25*x*BesselI(0, 2*x) - 6*(1-4*x)*BesselI(1, 2*x) ) - 9*exp(x) - 1/2. (End)

Extensions

More terms added by G. C. Greubel, Aug 22 2025