A046825 -id:A046825 - cp's OEIS frontend

A100513 Denominator of Sum_{k=0..n} 1/C(2n,2k).

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

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, ...} = A100512(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..1000

Cf. A046825, A100512, A100514, A100515.

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

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 *)

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

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

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

A100517 Denominator of Sum_{k=0..n} 1/binomial(n,k)^2.

Original entry on oeis.org

1, 1, 4, 9, 72, 10, 3600, 1575, 2800, 1764, 14112, 13475, 34927200, 2316600, 192192, 4459455, 4994589600, 262061800, 735869534400, 17476901442, 422721728, 353723760, 31127690880, 10150725585, 59637542956992, 2205530434800, 155748568976000, 50956005028500

Offset: 0

N. J. A. Sloane, Nov 25 2004

			1, 2, 9/4, 20/9, 155/72, 21/10, 7441/3600, 3224/1575, 5697/2800, 3575/1764, 28523/14112, 27183/13475, 70357417/34927200, 4661447/2316600, ... = A100516/A100517

H. W. Gould, Combinatorial Identities, Morgantown, 1972, p. 50, formula (5.2).

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

Cf. A000108, A046825, A046826, A100516, A100518, A100519.

[Denominator( (&+[1/Binomial(n,k)^2: k in [0..n]]) ): n in [0..40]]; // G. C. Greubel, Jun 24 2022

Table[Sum[1/Binomial[n,k]^2,{k,0,n}],{n,0,30}]//Denominator (* Harvey P. Dale, Apr 01 2019 *)

a(n) = denominator(sum(k=0, n, 1/binomial(n,k)^2)); \\ Michel Marcus, Jun 24 2022

[denominator(sum(1/binomial(n,k)^2 for k in (0..n))) for n in (0..40)] # G. C. Greubel, Jun 24 2022

a(n) = denominator( 3*(n+1)^2/((n+2)*(2*n+3)*Catalan(n+1)) * Sum_{k=1..n+1} binomial(2*k, k)/k ). - G. C. Greubel, Jun 24 2022

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

A100519 Denominator of Sum_{k=0..n} 1/binomial(n,k)^3.

Original entry on oeis.org

1, 1, 8, 27, 864, 500, 43200, 1157625, 42875, 571536, 8001504000, 61631955, 10650001824000, 8526987612000, 13865513485824, 91398648466125, 83564478597600, 4927753743913000, 4421332282230864000, 98559233902419862572, 340556687709473664000

Offset: 0

N. J. A. Sloane, Nov 25 2004

			1, 2, 17/8, 56/27, 1759/864, 1009/500, 86831/43200, 2322304/1157625, 85922/42875, 1144667/571536, 16019198113/8001504000, 123357293/61631955, ... = A100518/A100519.

G. C. Greubel, Table of n, a(n) for n = 0..750

Cf. A046825, A100516, A100517, A100518.

[Denominator( (&+[1/Binomial(n,k)^3: k in [0..n]]) ): n in [0..30]]; // G. C. Greubel, Jun 24 2022

Table[Denominator[Sum[1/Binomial[n,k]^3, {k,0,n}]], {n,0,30}] (* G. C. Greubel, Jun 24 2022 *)

a(n) = denominator(sum(k=0, n, 1/binomial(n,k)^3)); \\ Michel Marcus, Jun 25 2022

[denominator(sum(1/binomial(n,k)^3 for k in (0..n))) for n in (0..30)] # G. C. Greubel, Jun 24 2022

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

A212045 Numerators in the resistance triangle: T(k,n)=b, where b/c is the resistance distance R(k,n) for k resistors in an n-dimensional cube.

Original entry on oeis.org

1, 3, 1, 7, 3, 5, 15, 7, 61, 2, 31, 15, 241, 25, 8, 21, 31, 131, 101, 137, 13, 127, 21, 12, 7, 2381, 343, 151, 255, 127, 2105, 167, 10781, 2033, 32663, 32, 511, 255, 16531, 929, 42061, 9383, 84677, 2357, 83, 1023, 511, 5231, 7387, 74189, 1771, 12419

Offset: 1

Peter J. C. Moses, Apr 28 2012

The term "resistance distance" for electric circuits was in use years before it was proved to be a metric (on edges of graphs). The historical meaning has been described thus: "one imagines unit resistors on each edge of a graph G and takes the resistance distance between vertices i and j of G to be the effective resistance between vertices i and j..."  (from Klein, 2002; see the References).  Let R(k,n) denote the resistance distance for k resistors in an n-dimensional cube (for details, see Example and References).  Then
R(k,n)=A212045(k,n)/A212046(k,n).  Moreover,
A212045(1,n)=A090633(n), A212045(n,n)=A046878(n),
A212046(1,n)=A090634(n), A212046(n,n)=A046879(n).

			First six rows of A212045/A212046:
1
3/4 .... 1
7/12 ... 3/4 .... 5/6
15/32 .. 7/12 ... 61/96 ... 2/3
31/80 .. 15/32 .. 241/480 . 25/48 ... 8/15
21/64 .. 31/80 .. 131/320 . 101/240 . 137/320 . 13/30
The resistance distances for n=3 (the ordinary cube) are 7/12, 3/4, and 5/6, so that row 3 of the triangle of numerators is (7, 3, 5).  For the corresponding electric circuit, suppose X is a vertex of the cube. The resistance across any one of the 3 edges from X is 7/12 ohm; the resistance across any two adjoined edges (i.e., a diagonal of a face of the cubes) is 3/4 ohm; the resistance across and three adjoined edges (a diagonal of the cube) is 5/6 ohm.

F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.

D. J. Klein, Resistance Distance, Journal of Mathematical Chemistry 12 (1993) 81-95.
D. J. Klein, Resistance-Distance Sum Rules, Croatia Chemica Acta, 75 (2002), 633-649.
Nicholas Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, arXiv:0904.1757v1 [math.CO], 2009.
N. Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, Mathematics Magazine, 83:5 (2010) 331-346.
David Randall, The Resistor Cube Problem.
D. Singmaster, Problem 79-16, Resistances in an n-Dimensional Cube, SIAM Review, 22 (1980) 504.

Cf. A212046, A046878, A046879, A046825, A090634, A090633.

R[0, n_] := 0; R[1, n_] := (2 - 2^(1 - n))/n;
R[k_, n_] := R[k, n] = ((k - 1) R[k - 2, n] - n R[k - 1, n] + 2^(1 - n))/(k - n - 1)
t = Table[R[k, n], {n, 1, 11}, {k, 1, n}]
Flatten[Numerator[t]]    (* A212045 *)
Flatten[Denominator[t]]  (* A212046 *)
TableForm[Numerator[t]]
TableForm[Denominator[t]]

A212045(n)/A212046(n) is the rational number R(k, n) =

[(k-1)*R(k-2,n)-n*R(k-1,n)+2^(1-n)]/(k-n-1), for n>=1, k>=1.

A212046 Denominators in the resistance triangle: T(k,n)=b, where b/c is the resistance distance R(k,n) for k resistors in an n-dimensional cube.

Original entry on oeis.org

1, 4, 1, 12, 4, 6, 32, 12, 96, 3, 80, 32, 480, 48, 15, 64, 80, 320, 240, 320, 30, 448, 64, 35, 20, 6720, 960, 420, 1024, 448, 7168, 560, 35840, 6720, 107520, 105, 2304, 1024, 64512, 3584, 161280, 35840, 322560, 8960, 315, 5120, 2304, 23040, 32256

Offset: 1

Peter J. C. Moses, Apr 30 2012

The term "resistance distance" for electric circuits was in use years before it was proved to be a metric (on edges of graphs). The historical meaning has been described thus: "one imagines unit resistors on each edge of a graph G and takes the resistance distance between vertices i and j of G to be the effective resistance between vertices i and j..."  (from Klein, 2002; see the References). Let R(k,n) denote the resistance distance for k resistors in an n-dimensional cube (for details, see Example and References). Then
R(k,n)=A212045(k,n)/A212046(k,n).  Moreover,
A212045(1,n)=A090633(n), A212045(n,n)=A046878(n),
A212046(1,n)=A090634(n), A212046(n,n)=A046879(n).

			First six rows of A212045/A212046:
  1
  3/4 .... 1
  7/12 ... 3/4 .... 5/6
  15/32 .. 7/12 ... 61/96 ... 2/3
  31/80 .. 15/32 .. 241/480 . 25/48 ... 8/15
  21/64 .. 31/80 .. 131/320 . 101/240 . 137/320 . 13/30
The resistance distances for n=3 (the ordinary cube) are 7/12, 3/4, and 5/6, so that row 3 of the triangle of numerators is (7, 3, 5).  For the corresponding electric circuit, suppose X is a vertex of the cube.  The resistance across any one of the 3 edges from X is 7/12 ohm; the resistance across any two adjoined edges (i.e., a diagonal of a face of the cubes) is 3/4 ohm; the resistance across and three adjoined edges (a diagonal of the cube) is 5/6 ohm.

F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional cube, Quantum 7:1 (1996) 12-15 and 63.

D. J. Klein, Resistance Distance, Journal of Mathematical Chemistry 12 (1993) 81-95.
D. J. Klein, Resistance-Distance Sum Rules, Croatia Chemica Acta, Vol. 75, No. 2 (2002), 633-649.
Nicholas Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, arXiv:0904.1757 [math.CO], 2009.
N. Pippenger, The Hypercube of Resistors, Asymptotic Expansions, and Preferential Arrangements, Mathematics Magazine, 83:5 (2010) 331-346.
D. Singmaster, Problem 79-16, Resistances in an n-Dimensional Cube, SIAM Review, 22 (1980) 504.
Wikipedia, Resistance distance

Cf. A212045, A046878, A046879, A046825, A090634, A090633.

R[0, n_] := 0; R[1, n_] := (2 - 2^(1 - n))/n;
R[k_, n_] := R[k, n] = ((k - 1) R[k - 2, n] - n R[k - 1, n] + 2^(1 - n))/(k - n - 1)
t = Table[R[k, n], {n, 1, 11}, {k, 1, n}]
Flatten[Numerator[t]]    (* A212045 *)
Flatten[Denominator[t]]  (* A212046 *)
TableForm[Numerator[t]]
TableForm[Denominator[t]]

A212045(n)/A212046(n) is the rational number R(k, n) =

[(k-1)*R(k-2,n)-n*R(k-1,n)+2^(1-n)]/(k-n-1), for n>=1, k>=1.

A354478 a(n) is the numerator of Sum_{k=1..n} 1 / Stirling1(n,k).

Original entry on oeis.org

1, 0, 7, 25, 3991, 3923773, 4901627, 527165212865, 9823031039961293027, 123877274974851473572937, 443645907754951021537851199, 246932542361393897304051461727006396307, 1474846779473982897350113519971401527250089, 46578509609937575127608478711343978511593638945099881

Offset: 1

Ilya Gutkovskiy, Jun 02 2022

Conjecture: a(n)/A354479(n) tends to 1 as n tends to infinity. For comparison: A112290(n)/A112291(n) tends to 2 as n tends to infinity. - Vaclav Kotesovec, Jun 02 2022

			1, 0, 7/6, 25/33, 3991/4200, 3923773/4192200, 4901627/5115600, 527165212865/545250747888, ...

Cf. A008275, A046825, A112288, A112290, A354479 (denominators).

Table[Sum[1/StirlingS1[n, k], {k, 1, n}], {n, 1, 14}] // Numerator

a(n) = numerator(sum(k=1, n, 1/stirling(n, k, 1))); \\ Michel Marcus, Jun 02 2022

A122184 Numerator of Sum_{k=0..2n} (-1)^k/C(2n,k)^3.

Original entry on oeis.org

1, 15, 1705, 47789, 1369377, 213162301, 43005554527, 14505995375, 23869750002797, 2384790127843063, 624724994927411, 24386251366041479501, 2042595777439018142725, 11191251831905709132993

Offset: 0

Alexander Adamchuk, May 10 2007

p^k divides a((p^k+1)/2) for prime p>2 and integer k>0.

Eric Weisstein's World of Mathematics, Binomial Sums.

Cf. A046825 = Numerator of Sum_{k=0..n} 1/C(n, k). Cf. A100516 = Numerator of Sum_{k=0..n} 1/C(n, k)^2. Cf. A100518 = Numerator of Sum_{k=0..n} 1/C(n, k)^3. Cf. A100520 = Numerator of Sum_{k=0..2n} (-1)^k/C(2n, k)^2.

Table[ Numerator[ Sum[ (-1)^k / Binomial[2n,k]^3, {k,0,2n} ] ], {n,0,25} ]

a(n) = Numerator[ Sum[ (-1)^k / Binomial[2n,k]^3, {k,0,2n} ] ].

cp's OEIS Frontend

A100513 Denominator of Sum_{k=0..n} 1/C(2*n,2*k).

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A100517 Denominator of Sum_{k=0..n} 1/binomial(n,k)^2.

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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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A100519 Denominator of Sum_{k=0..n} 1/binomial(n,k)^3.

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A100513 Denominator of Sum_{k=0..n} 1/C(2n,2k).

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).