A100513
Denominator of Sum_{k=0..n} 1/C(2*n,2*k).
Original entry on oeis.org
1, 1, 6, 15, 35, 315, 13860, 3003, 9009, 765765, 1385670, 14549535, 66927861, 371821450, 40156716600, 145568097675, 136745788725, 128931743655, 9025222055850, 4281195077775, 166966608033225, 6845630929362225, 26165522663340060, 294362129962575675
Offset: 0
Sum_{k=0..n} 1/binomial(2*n,2*k) = {1, 2, 13/6, 32/15, 73/35, 647/315, 28211/13860, 6080/3003, 18181/9009, 1542158/765765, 2786599/1385670, 29229544/14549535, 134354573/66927861, ...} = A100512(n)/a(n).
- M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 126-127.
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[Denominator((&+[1/Binomial(2*n, 2*k): k in [0..n]])): n in [0..40]]; // G. C. Greubel, Mar 28 2023
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Table[Denominator[(2*n+1)*Sum[Beta[2k+1,2(n-k)+1], {k,0,n}]], {n,0,40}] (* G. C. Greubel, Mar 28 2023 *)
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def A100513(n): return denominator((2*n+1)*sum(beta(2*k+1, 2*(n-k)+1) for k in range(n+1)))
[A100513(n) for n in range(40)] # G. C. Greubel, Mar 28 2023
A100515
Denominator of Sum_{k=0..n} 1/C(3*n, 3*k).
Original entry on oeis.org
1, 1, 20, 42, 4620, 5005, 2042040, 1763580, 59491432, 95611230, 776363187600, 235953517800, 24067258815600, 143627189706, 12170010541088400, 34128942604356600, 245138783756209200, 1103648327722933300, 497725329469811592240, 94396183175309095080, 538372898043179538939600
Offset: 0
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).
- M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 126-127.
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[Denominator((&+[1/Binomial(3*n, 3*k): k in [0..n]])): n in [0..40]]; // G. C. Greubel, Mar 28 2023
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Table[Denominator[(3*n+1)*Sum[Beta[3k+1,3n-3k+1], {k,0,n}]], {n,0,40}] (* G. C. Greubel, Mar 28 2023 *)
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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
A100512
Numerator of Sum_{k=0..n} 1/C(2*n, 2*k).
Original entry on oeis.org
1, 2, 13, 32, 73, 647, 28211, 6080, 18181, 1542158, 2786599, 29229544, 134354573, 745984697, 80530073893, 291816652544, 274050911261, 258328905974, 18079412000719, 8574689239808, 334365081328507, 13707288497202919, 52386756782140399, 589296748617180608
Offset: 0
Sum_{k=0..n} 1/binomial(2*n, 2*k) = {1, 2, 13/6, 32/15, 73/35, 647/315, 28211/13860, 6080/3003, 18181/9009, 1542158/765765, 2786599/1385670, 29229544/14549535, 134354573/66927861, ...} = a(n)/A100513(n).
- M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. 126-127.
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[Numerator((&+[1/Binomial(2*n, 2*k): k in [0..n]])): n in [0..40]]; // G. C. Greubel, Mar 28 2023
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Table[Sum[1/Binomial[2n,2k],{k,0,n}],{n,0,30}]//Numerator (* Harvey P. Dale, Aug 12 2016 *)
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def A100512(n): return numerator((2*n+1)*sum(beta(2*k+1, 2*n-2*k+1) for k in range(n+1)))
[A100512(n) for n in range(40)] # G. C. Greubel, Mar 28 2023
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