A274104 a(n) = Sum_{k=0..n} (3*k+2)*Catalan(k).

2, 7, 23, 78, 274, 988, 3628, 13495, 50675, 191673, 729145, 2786655, 10691111, 41150011, 158825371, 614483086, 2382366586, 9253540456, 36001307656, 140269835866, 547245301906, 2137552658206, 8358366985726, 32715599554876, 128168506456852, 502538379368656, 1971926625140816

Offset: 0

Author

N. J. A. Sloane, Jun 13 2016

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Moa Apagodu and Doron Zeilberger, Using the "Freshman's Dream" to Prove Combinatorial Congruences, arXiv:1606.03351 [math.CO], 2016. Also Amer. Math. Monthly. 124 (2017), 597-608.

Crossrefs

Partial sums of A051960.

Cf. A000108, A006134, A014137.

Programs

Magma

[(&+[(3*k+2)*Catalan(k): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Jun 30 2024

Mathematica

CoefficientList[Series[(1 +2 x -Sqrt[1-4 x])/(2 x Sqrt[1-4 x] (1-x)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 18 2016 *)

SageMath

[sum((3*k+2)*catalan_number(k) for k in range(n+1)) for n in range(41)] # G. C. Greubel, Jun 30 2024

Formula

D-finite with recurrence: (n+1)*a(n) - (3*n+5)*a(n-1) - 2*(3*n-8)*a(n-2) + 4*(2*n-3)*a(n-3) = 0. - R. J. Mathar, Jun 15 2016

G.f.: (1 + 2*x - sqrt(1-4*x))/(2*x*(1-x)*sqrt(1-4*x)). - Ilya Gutkovskiy, Jun 15 2016

a(n) = A014137(n+1) + (n+1)*A000108(n+1) - 1. - G. C. Greubel, Jun 30 2024

From Mélika Tebni, Sep 02 2024: (Start)

a(n) = A006134(n) + A006134(n+1)/2 - 1/2.

E.g.f.: exp(2*x)*(5*BesselI(0, 2*x)/2 + BesselI(1, 2*x)) + exp(x)/2*(3*Integral_{x=-oo..oo} BesselI(0,2*x)*exp(x) dx - 1). (End)