A100511 a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n,j)*binomial(n,k)*max(j,k).

0, 3, 22, 126, 652, 3190, 15060, 69356, 313624, 1398438, 6166660, 26948548, 116888232, 503811516, 2159864392, 9216445080, 39168381488, 165864540934, 700151508324, 2947120122068, 12373581565960, 51831196048212, 216659135089496, 903925011410536

Offset: 0

Author

N. J. A. Sloane, Nov 24 2004

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 127-129.

Crossrefs

Cf. A000108, A002457, A033504.

Programs

Magma

[n*(4^n +(n+1)*Catalan(n))/2: n in [0..40]]; // G. C. Greubel, Apr 01 2023

Mathematica

Table[n*(4^n +(n+1)*CatalanNumber[n])/2, {n,0,40}] (* G. C. Greubel, Apr 01 2023 *)

PARI

a(n) = n*2^(2*n-1) + (n/2)*binomial(2*n, n); \\ Michel Marcus, Dec 26 2017

SageMath

[n*(4^n +binomial(2*n,n))/2 for n in range(41)] # G. C. Greubel, Apr 01 2023

Formula

a(n) = n*2^(2*n-1) + (n/2)*binomial(2*n, n). [Typo corrected by Ognjen Dragoljevic, Dec 26 2017]

From G. C. Greubel, Apr 01 2023: (Start)

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

E.g.f.: x*(2*exp(4*x)+ exp(2*x)*(BesselI(0, 2*x) + BesselI(1, 2*x))). (End)