A268590 -id:A268590 - cp's OEIS frontend

A034602 Wolstenholme quotient W_p = (binomial(2p-1,p) - 1)/p^3 for prime p=A000040(n).

1, 5, 265, 2367, 237493, 2576561, 338350897, 616410400171, 7811559753873, 17236200860123055, 3081677433937346539, 41741941495866750557, 7829195555633964779233, 21066131970056662377432067, 59296957594629000880904587621, 844326030443651782154010715715

Offset: 3

N. J. A. Sloane

nonn

Equivalently, (binomial(2p,p)-2)/(2*p^3) where p runs through the primes >=5.

The values of this sequence's terms are replicated by conjectured general formula, given in A223886 (and also added to the formula section here) for k=2, j=1 and n>=3. - Alexander R. Povolotsky, Apr 18 2013

			Binomial(10,5)-2 = 250; 5^3=125 hence a(5)=1.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B31.

Robert Israel, Table of n, a(n) for n = 3..263
R. R. Aidagulov and M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0 arXiv:1602.02632
R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica 71:4 (1995), 381-389.
Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Jonathan Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; arXiv:1812.07650 [math.NT], 2018.

Cf. A177783 (alternative definition of Wolstenholme quotient), A072984, A092101, A092103, A092193, A128673, A217772, A223886, A263882.

Cf. A268512, A268589, A268590.

[(Binomial(2*p-1,p)-1) div p^3: p in PrimesInInterval(4,100)]; // Vincenzo Librandi, Nov 23 2015

f:= proc(n) local p;
p:= ithprime(n);
(binomial(2*p-1,p)-1)/p^3
end proc:
map(f, [$3..30]); # Robert Israel, Dec 19 2018

Table[(Binomial[2 Prime[n] - 1, Prime[n] - 1] - 1)/Prime[n]^3, {n, 3, 20}] (* Vincenzo Librandi, Nov 23 2015 *)

a(n) = (A088218(p)-1)/p^3 = (A001700(p-1)-1)/p^3 = (A000984(p)-2)/(2*p^3), where p=A000040(n).
a(n) = A087754(n)/2.
a(n) = (binomial(j*k*prime(n), j*prime(n)) - binomial(k*j, j)) / (k*prime(n)^3) for k=2, j=1, and n>=3. - Alexander R. Povolotsky, Apr 18 2013
a(n) = A263882(n)/prime(n) for n > 2. - Jonathan Sondow, Nov 23 2015
a(n) = numerator(tanh(Sum_{k=1..p-1} artanh(k/p)))/p^3, where p = prime(n) for n >= 3. - Thomas Ordowski, Apr 17 2025

Edited by Max Alekseyev, May 14 2010

More terms from Vincenzo Librandi, Nov 23 2015

A268589 a(n) = (2C(3p,p) - 9C(2p,p) + 12) / p^5, where p = prime(n).

Original entry on oeis.org

12, 2364, 43500, 20791626, 514377588, 373783661124, 9888937247184828, 312285010312512084, 11167980739981519994382, 13185583459205473525798038, 462369843775374621687338484, 588608385261717115044847555476, 28758863221144089886068560242560564, 1508365481231852329668720928730586740868

Offset: 4

Max Alekseyev, Feb 07 2016

a(n) is an integer for all n>=4, see A268512.

R. R. Aidagulov and M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0; arXiv preprint, arXiv:1602.02632 [math.NT], 2016-2018.

Cf. A268512, A087754, A268590.

Table[(2*Binomial[3p,p]-9*Binomial[2p,p]+12)/p^5,{p,Prime[Range[4,20]]}] (* Harvey P. Dale, Aug 27 2025 *)

{ A268589(n) = my(p=prime(n)); (12 - 9*binomial(2*p,p) + 2*binomial(3*p,p))/p^5; }

A087754 a(n) = (C(2p,p)-2) / p^3, where p = prime(n).

Original entry on oeis.org

2, 10, 530, 4734, 474986, 5153122, 676701794, 1232820800342, 15623119507746, 34472401720246110, 6163354867874693078, 83483882991733501114, 15658391111267929558466, 42132263940113324754864134

Offset: 3

Henry Bottomley, Oct 02 2003

nonn

			a(6)=4734 since 13 is the sixth prime and (C(26,13)-2)/13^3 = (10400600-2)/2197 = 4734.

R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0 arXiv:1602.02632
R. P. Stanley, MIT Course OCW 18.S66, The Art of Counting, Spring 2003. (Problem 18)

Cf. A268512, A268589, A268590.

Table[(Binomial[2p,p]-2)/p^3,{p,Prime[Range[3,20]]}] (* Harvey P. Dale, Oct 23 2017 *)

a(n) = A060842(n) / A000040(n).

a(n) = 2 * A034602(n).

A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^ic(n,i)binomial(i*p,p) is divisible by p^(2n-1).

Original entry on oeis.org

1, 2, 1, 12, 9, 2, 60, 54, 20, 3, 840, 840, 400, 105, 12, 2520, 2700, 1500, 525, 108, 10, 27720, 31185, 19250, 8085, 2268, 385, 30, 360360, 420420, 280280, 133770, 45864, 10780, 1560, 105, 720720, 864864, 611520, 321048, 127008, 36960, 7488, 945, 56, 12252240, 15036840, 11138400, 6297480, 2776032, 942480

Offset: 1

Max Alekseyev, Feb 06 2016

			n=1: 1
n=2: 2, 1
n=3: 12, 9, 2
n=4: 60, 54, 20, 3
n=5: 840, 840, 400, 105, 12
...
For all primes p>3, p^3 divides 2 - binomial(2*p,p) (cf. A087754).
For all primes p>5, p^5 divides 12 - 9*binomial(2*p,p) + 2*binomial(3*p,p) (cf. A268589).
For all primes p>7, p^7 divides 60 - 54*binomial(2*p,p) + 20*binomial(3*p,p) - 3*binomial(4*p,p) (cf. A268590).

R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0; also arXiv, arXiv:1602.02632 [math.NT], 2016-2018.

Cf. A099996 (first column), A068550 (diagonal), A087754, A268589, A268590, A254593.

a3418[n_] := LCM @@ Range[n];
c[1, 1] = 1; c[n_, i_] := a3418[2(n-1)] Binomial[2n-1, n-i] ((2i-1)/i/ Binomial[2n-1, n]);
Table[c[n, i], {n, 1, 10}, {i, 1, n}] // Flatten (* Jean-François Alcover, Dec 04 2018 *)

{ A268512(n,i) = lcm(vector(2*(n-1),i,i)) * binomial(2*n-1,n-i) * (2*i-1) / i / binomial(2*n-1,n) }

c(n,i) = A003418(2*(n-1))*binomial(2*n-1,n-i)*(2*i-1)/i/binomial(2*n-1,n).

A357955 a(n) = 3binomial(4n,n) - 20binomial(3n,n) + 54binomial(2n,n).

Original entry on oeis.org

37, 60, 108, 60, -660, 60, 82404, 1411848, 17540460, 191318820, 1952058108, 19175376324, 184118073828, 1743153802320, 16359157606200, 152693295412560, 1420516291306860, 13190159377278324, 122358232382484420, 1134645084249344400, 10522118980232969340

Offset: 0

Peter Bala, Oct 22 2022

Conjectures:
1) a(p) == a(1) (mod p^7) for all primes p >= 3 except p = 7.
2) a(p^r) == a(p^(r-1)) ( mod p^(3*r+5) ) for r >= 2 and all primes p >= 3.
These are stronger supercongruences than those satisfied separately by the sequences {binomial(2*n,n)} = A000984, {binomial(3*n,n)} = A005809 and {binomial(4*n,n)} = A005810.
Conjecture 1) was proved by Aidagulov and Alekseyev; see the remarks following Corollary 2. - Peter Bala, Oct 29 2022

			Examples of supercongruences:
a(11) - a(1) = 19175376324 - 60 = (2^3)*3*(11^7)*41 == 0 (mod 11^7).
a(5^2) - a(5) = 726506045044361132812560 - 60 = (2^2)*3*(5^11)*41*30241552444123 == 0 (mod 5^11).

Paolo Xausa, Table of n, a(n) for n = 0..1000
R. R. Aidagulov and M. A. Alekseyev, On p-adic approximation of sums of binomial coefficients, Journal of Mathematical Sciences 233:5 (2018), 626-634; arXiv:1602.02632 [math.NT], 2018.
C. Helou and G. Terjanian, On Wolstenholme’s theorem and its converse, J. Number Theory 128 (2008), 475-499.
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv preprint arXiv:1111.3057 [math.NT], 2011.

Cf. A000984, A005809, A005810, A268590, A357509, A357567, A357568, A357569.

seq(3*binomial(4*n,n) - 20*binomial(3*n,n) + 54*binomial(2*n,n), n = 0..20);

A357955[n_] := 3*Binomial[4*n, n] - 20*Binomial[3*n, n] + 54*Binomial[2*n, n];
Array[A357955, 25, 0] (* Paolo Xausa, Jul 17 2024 *)

from math import comb
def A357955(n): return 54*comb(m:=n<<1,n)+3*comb(m<<1,n)-20*comb(m+n,n) # Chai Wah Wu, Oct 24 2022

a(n) = 3*A005810(n) - 20*A005809(n) + 54*A000984(n).
a(k*p^r) == a(k*p^(r-1)) ( mod p^(3*r) ) for positive integers k and r and for all primes p >= 5 (see Meštrović, Section 6, equation 39).
a(p) == a(1) (mod p^6) for all primes p >= 7 (apply Helou and Terjanian, Section 3, Proposition 2).
From Stefano Spezia, Jul 17 2024: (Start)
G.f.: 54/sqrt(1-4*x) - 40*cos(arccos(1-27*x/2)/6)/sqrt(4-27*x) + 3*hypergeom([1/4, 1/2, 3/4], [1/3, 2/3], 4^4*x/3^3).
E.g.f.: 54*exp(2*x)*BesselI(0, 2*x) - 20*hypergeom([1/3, 2/3], [1/2, 1], 27*x/4) + 3*hypergeom([1/4, 1/2, 3/4], [1/3, 2/3, 1], 4^4*x/3^3).
a(n) ~ exp(2*n*arctanh(229/283))*sqrt(6/(n*Pi)). (End)

cp's OEIS Frontend

A034602 Wolstenholme quotient W_p = (binomial(2p-1,p) - 1)/p^3 for prime p=A000040(n).

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A268589 a(n) = (2*C(3p,p) - 9*C(2p,p) + 12) / p^5, where p = prime(n).

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A087754 a(n) = (C(2p,p)-2) / p^3, where p = prime(n).

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A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^i*c(n,i)*binomial(i*p,p) is divisible by p^(2n-1).

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Programs

Mathematica

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Formula

A357955 a(n) = 3*binomial(4*n,n) - 20*binomial(3*n,n) + 54*binomial(2*n,n).

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Comments

Examples

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Maple

Mathematica

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Formula

A268589 a(n) = (2C(3p,p) - 9C(2p,p) + 12) / p^5, where p = prime(n).

A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^ic(n,i)binomial(i*p,p) is divisible by p^(2n-1).

A357955 a(n) = 3binomial(4n,n) - 20binomial(3n,n) + 54binomial(2n,n).