A100514 - OEIS

%I #7 Mar 31 2023 05:12:57

%S 1,2,41,85,9287,10034,4089347,3529889,119042647,191288533,

%T 1553111566613,471993968921,48141284433673,287285900609,

%U 24342145990117741,68262703949495173,490305954062679017,2207402771385797549,995490830339080453219,188798823808438240073

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

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

%H G. C. Greubel, <a href="/A100514/b100514.txt">Table of n, a(n) for n = 0..765</a>

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

%F a(n) = numerator( (3*n+1)*Sum_{k=0..n} beta(3*k+1, 3*(n-k)+1) ). - _G. C. Greubel_, Mar 28 2023

%e 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).

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

%o (Magma) [Numerator((&+[1/Binomial(3*n, 3*k): k in [0..n]])): n in [0..40]]; // _G. C. Greubel_, Mar 28 2023

%o (SageMath)

%o def A100514(n): return numerator((3*n+1)*sum(beta(3*k+1, 3*n-3*k+1) for k in range(n+1)))

%o [A100514(n) for n in range(40)] # _G. C. Greubel_, Mar 28 2023

%Y Cf. A046825, A100513, A100515.

%K nonn,frac

%O 0,2

%A _N. J. A. Sloane_, Nov 25 2004