A100515 Denominator of Sum_{k=0..n} 1/C(3*n, 3*k).

1, 1, 20, 42, 4620, 5005, 2042040, 1763580, 59491432, 95611230, 776363187600, 235953517800, 24067258815600, 143627189706, 12170010541088400, 34128942604356600, 245138783756209200, 1103648327722933300, 497725329469811592240, 94396183175309095080, 538372898043179538939600

Offset

0,3

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

Formula

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

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

Example

Sum_{k=0..n} 1/binomial(3*n,3*k) = { 1, 2, 41/20, 85/42, 9287/4620, 10034/5005, 4089347/2042040, 3529889/1763580, 119042647/59491432, 191288533/95611230, 1553111566613/776363187600, ...} = A100514(n)/a(n).

Mathematica

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

Prog

(Magma) [Denominator((&+[1/Binomial(3*n, 3*k): k in [0..n]])): n in [0..40]]; // G. C. Greubel, Mar 28 2023
(SageMath)
def A100515(n): return denominator((3*n+1)*sum(beta(3*k+1, 3*n-3*k+1) for k in range(n+1)))
[A100515(n) for n in range(40)] # G. C. Greubel, Mar 28 2023

Crossrefs

Cf. A046825, A100513, A100514.

Sequence in context: A128672 A290184 A126251 * A220006 A295795 A039322

Adjacent sequences: A100512 A100513 A100514 * A100516 A100517 A100518

Keyword

nonn,frac

Author

N. J. A. Sloane, Nov 25 2004

Status

approved