A100514 Numerator of Sum_{k=0..n} 1/C(3*n, 3*k).

1, 2, 41, 85, 9287, 10034, 4089347, 3529889, 119042647, 191288533, 1553111566613, 471993968921, 48141284433673, 287285900609, 24342145990117741, 68262703949495173, 490305954062679017, 2207402771385797549, 995490830339080453219, 188798823808438240073

Offset: 0

Author

N. J. A. Sloane, Nov 25 2004

Keywords

• nonn
• frac

Examples

			Sum_{k=0..n} 1/C(3*n, 3*k) = { 1, 2, 41/20, 85/42, 9287/4620, 10034/5005, 4089347/2042040, 3529889/1763580, 119042647/59491432, 191288533/95611230, 1553111566613/776363187600, ...} = a(n)/A100515(n).
		

References

• M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 126-127.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..765

Crossrefs

Cf. A046825, A100513, A100515.

Programs

Magma

[Numerator((&+[1/Binomial(3*n, 3*k): k in [0..n]])): n in [0..40]]; // G. C. Greubel, Mar 28 2023

Mathematica

Table[Numerator[(3*n+1)*Sum[Beta[3k+1,3n-3k+1], {k,0,n}]], {n,0,40}] (* G. C. Greubel, Mar 28 2023 *)

SageMath

def A100514(n): return numerator((3*n+1)*sum(beta(3*k+1, 3*n-3*k+1) for k in range(n+1)))
[A100514(n) for n in range(40)] # G. C. Greubel, Mar 28 2023

Formula

a(n) = numerator( Sum_{k=0..n} 1/C(3*n, 3*k) ).

a(n) = numerator( (3*n+1)*Sum_{k=0..n} beta(3*k+1, 3*(n-k)+1) ). - G. C. Greubel, Mar 28 2023