A100513 -id:A100513 - cp's OEIS frontend

A046825 Numerator of Sum_{k=0..n} 1/binomial(n,k).

1, 2, 5, 8, 8, 13, 151, 256, 83, 146, 1433, 2588, 15341, 28211, 52235, 19456, 19345, 36362, 651745, 6168632, 1463914, 2786599, 122289917, 233836352, 140001721, 268709146, 774885169, 1491969394, 41711914513, 80530073893

Offset: 0

N. J. A. Sloane

The term a(12)=15341 is divisible by 23^2. Is there another term a(n) divisible by the square of a prime p larger than n+1? - M. F. Hasler, Jul 17 2012

			1, 2, 5/2, 8/3, 8/3, 13/5, 151/60, 256/105, 83/35, 146/63, 1433/630, 2588/1155, 15341/6930, 28211/12870, 52235/24024, 19456/9009, 19345/9009, ... = A046825/A046826

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294, Problem 7.15.
R. L. Graham, D. E. Knuth, O. Patashnik; Concrete Mathematics, Addison-Wesley, Reading (1994) 2nd Ed. Exercise 5.100.
G. Letac, Problèmes de probabilités, Presses Universitaires de France (1970), p. 14.
F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.

T. D. Noe, Table of n, a(n) for n = 0..200
T. Mansour, Gamma function, beta function and combinatorial identities.
T. Sillke, More information
D. Singmaster, Problem 79-16, Resistances in an n-Dimensional Cube, SIAM Review, 22 (1980) 504.
B. Sury, Sum of the reciprocals of the binomial coefficients, Europ. J. Combinatorics, 14 (1993), 351-353.

Cf. A003149, A046826, A048211, A051389, A100512, A100513.

[Numerator((&+[1/Binomial(n,j): j in [0..n]])): n in [0..40]]; // G. C. Greubel, May 24 2021

Numerator/@Table[Sum[1/Binomial[n,k],{k,0,n}],{n,0,40}]  (* Harvey P. Dale, Apr 21 2011 *)

P=1;vector(30,n,numerator(P)+0*P=P/2/n*(n+1)+1) \\ M. F. Hasler, Jul 17 2012

A046825(n)=numerator(sum(k=0,n,1/binomial(n,k))) \\ M. F. Hasler, Jul 19 2012

[numerator(sum(1/binomial(n,j) for j in (0..n))) for n in (0..40)] # G. C. Greubel, May 24 2021

Let P(n) = (1/n) * Sum_{k=0..n-1} 1/binomial(n-1, k) = A046878(n)/A046879(n) = A046825(n-1)/(n*A046826(n-1)): { 0, 1, 1, 5/6, 2/3, 8/15, ...}. Then P(n) = 2^(-n) * Sum_{k=1..n} 2^k / k = 2^(-n+1) * Sum_{k odd} binomial(n, k)/k; P(0) = 0, P(n) = P(n-1)/2 + 1/n. - Torsten Sillke (Torsten.Sillke(AT)uni-bielefeld.de)
G.f. for P(n): (2*log(1-z))/(-2+z). - Wouter Meeussen
P(n) = 2^(-n) * Sum_{k=1..n} (binomial(n,k) + 1)/k.
a(n) = numerator( A003149(n)/n! ). - G. C. Greubel, May 24 2021

References entries (Comtet, Graham et al., Letac, Nedemeyer) and Links entries (Singmaster, Sury) from Torsten.Sillke(AT)uni-bielefeld.de

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

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Nov 25 2004

			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).

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

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

Cf. A046825, A100513, A100515.

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

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

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

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

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

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

N. J. A. Sloane, Nov 25 2004

			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.

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

Cf. A046825, A100513, A100514.

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

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

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

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

A100512 Numerator of Sum_{k=0..n} 1/C(2n, 2k).

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

N. J. A. Sloane, Nov 25 2004

			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.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A046825, A100513, A100514, A100515.

[Numerator((&+[1/Binomial(2*n, 2*k): k in [0..n]])): n in [0..40]]; // G. C. Greubel, Mar 28 2023

Table[Sum[1/Binomial[2n,2k],{k,0,n}],{n,0,30}]//Numerator (* Harvey P. Dale, Aug 12 2016 *)

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

a(n) = numerator( Sum_{k=0..n} 1/binomial(2*n, 2*k) ).

a(n) = numerator( (2*n+1)*Sum_{k=0..n} beta(2*k+1, 2*n-2*k+1) ). - G. C. Greubel, Mar 28 2023

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A046825 Numerator of Sum_{k=0..n} 1/binomial(n,k).

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

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

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A100512 Numerator of Sum_{k=0..n} 1/C(2*n, 2*k).

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

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

A100512 Numerator of Sum_{k=0..n} 1/C(2n, 2k).