A260793 -id:A260793 - cp's OEIS frontend

A290575 Apéry-like numbers Sum_{k=0..n} (C(n,k) * C(2*k,n))^2.

1, 4, 40, 544, 8536, 145504, 2618176, 48943360, 941244376, 18502137184, 370091343040, 7508629231360, 154145664817600, 3196100636757760, 66834662101834240, 1407913577733228544, 29849617614785770456, 636440695668355742560, 13638210075999240396736, 293565508750164008207104, 6344596821114216520841536

Offset: 0

Hugo Pfoertner, Aug 06 2017

Sequence epsilon in Almkvist, Straten, Zudilin article.

Seiichi Manyama, Table of n, a(n) for n = 0..500
G. Almkvist, D. van Straten, and W. Zudilin, Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinburgh Math. Soc.54 (2) (2011), 273-295.
Shaun Cooper, Apéry-like sequences defined by four-term recurrence relations, arXiv:2302.00757 [math.NT], 2023.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See epsilon p. 3.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
Armin Straub and Wadim Zudilin, Positivity of rational functions and their diagonals, J. Approx. Theory, 195:57-69, 2015. arXiv:1312.3732 [math.NT].
Zhi-Hong Sun, Congruences for Apéry-like numbers, arXiv:1803.10051 [math.NT], 2018.
Zhi-Hong Sun, New congruences involving Apéry-like numbers, arXiv:2004.07172 [math.NT], 2020.
Jeremy Tan, Simplifying a binomial sum for bridge deals with specific voids, Mathematics Stack Exchange, 2024.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

Table[Sum[(Binomial[n, k]*Binomial[2*k, n])^2, {k, 0, n}], {n, 0, 25}] (* G. C. Greubel, Oct 23 2017 *)
a[n_] := Binomial[2 n, n]^2 HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1, 1/2 - n, 1/2 - n}, 1];
Table[a[n], {n, 0, 20}] (* Peter Luschny, Apr 10 2022 *)

C=binomial; a(n) = sum (k=0, n, C(n,k)^2 * C(k+k,n)^2);

a(-1)=0, a(0)=1, a(n+1) = ((2*n+1)*(12*n^2+12*n+4)*a(n)-16*n^3*a(n-1))/(n+1)^3.
a(n) = Sum_{k=ceiling(n/2)..n} binomial(n,k)^2*binomial(2*k,n)^2. [Gorodetsky] - Michel Marcus, Feb 25 2021
a(n) ~ 2^(2*n - 3/4) * (1 + sqrt(2))^(2*n+1) / (Pi*n)^(3/2). - Vaclav Kotesovec, Jul 10 2021
From Peter Bala, Apr 10 2022: (Start)
The g.f. is the diagonal of the rational function 1/(1 - (x + y + z + t) + 2*(x*y*z + x*y*t + x*z*t + y*z*t) + 4*x*y*z*t) (Straub and Zudilin)
The g.f. appears to be the diagonal of the rational function 1/(1 - x - y + z - t - 2*(x*z + y*z + z*t) + 4*(x*y*t + x*z*t) + 8*x*y*z*t).
If true, then a(n) = [(x*y*z)^n] ( (x + y + z + 1)*(x + y + z - 1)*(x + y - z - 1)*(x - y - z + 1) )^n . (End)
a(n) = binomial(2*n, n)^2 * hypergeom([1/2-n/2, 1/2-n/2, -n/2, -n/2], [1, 1/2-n, 1/2-n], 1). - Peter Luschny, Apr 10 2022
G.f.: hypergeom([1/8, 3/8],[1], 256*x^2 / (1 - 4*x)^4)^2 / (1 - 4*x). - Mark van Hoeij, Nov 12 2022
a(n) = [(w*x*y*z)^n] ((w+z)*(x+z)*(y+z)*(w+x+y+z))^n = Sum_{0 <= j <= i <= n} binomial(n,i)^2*binomial(i,j)^2*binomial(n+j,i). - Jeremy Tan, Mar 28 2024

A290576 Apéry-like numbers Sum_{k=0..n} Sum_{l=0..n} (C(n,k)^2C(n,l)C(k,l)*C(k+l,n)).

Original entry on oeis.org

1, 3, 27, 309, 4059, 57753, 866349, 13492251, 216077787, 3536145057, 58875891777, 994150929951, 16984143140589, 293036113226223, 5098773125244483, 89368239352074309, 1576424378494272987, 27964450505226314673, 498550055166916502121

Offset: 0

Hugo Pfoertner, Aug 06 2017

Sequence zeta (formula 4.12) in Almkvist, Straten, Zudilin article.

Robert Israel, Table of n, a(n) for n = 0..779
G. Almkvist, D. van Straten, and W. Zudilin, Generalizations of Clausen’s formula and algebraic transformations of Calabi-Yau differential equations, Proc. Edinburgh Math. Soc.54 (2) (2011), 273-295.
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See zeta p. 3.
Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

f:= gfun:-rectoproc({a(0)=1, a(1)=3, a(n+1) = ( (2*n+1)*(9*n^2+9*n+3)*a(n) + 27*n^3*a(n-1) ) / (n+1)^3}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Aug 07 2017

Table[Sum[Sum[(Binomial[n, k]^2*Binomial[n, j] Binomial[k, j] Binomial[k + j, n]), {j, 0, n} ], {k, 0, n}], {n, 0, 18}] (* Michael De Vlieger, Aug 07 2017 *)

C=binomial;
a(n) = sum(k=0,n, sum(l=0,n, C(n,k)^2 * C(n,l) * C(k,l) * C(k+l,n) ));

a(0) = 1, a(1) = 3,
a(n+1) = ( (2*n+1)*(9*n^2+9*n+3)*a(n) + 27*n^3*a(n-1) ) / (n+1)^3.
a(n) ~ 3^(3*n/2 + 1) * (1+sqrt(3))^(2*n+1) / (2^(n + 5/2) * (Pi*n)^(3/2)). - Vaclav Kotesovec, Jul 10 2021
G.f.: hypergeom([1/12,5/12],[1],(12*x/(1-6*x-27*x^2))^3)^2/(1-6*x-27*x^2)^(1/2). - Mark van Hoeij, Nov 11 2022

A133370 Primes p such that p does not divide any term of the Apery sequence A005259 .

Original entry on oeis.org

2, 3, 7, 13, 23, 29, 43, 47, 53, 67, 71, 79, 83, 89, 101, 103, 107, 109, 113, 127, 131, 137, 149, 157, 167, 173, 199, 223, 229, 239, 263, 269, 277, 281, 311, 313, 317, 337, 349, 353, 359, 373, 383, 389, 397, 401, 409, 421, 449, 457, 461, 467, 479, 487, 491

Offset: 1

Philippe Deléham, Oct 27 2007

nonn

Malik and Straub give arguments suggesting that this sequence is infinite. - N. J. A. Sloane, Aug 06 2017

Robert Price, Table of n, a(n) for n = 1..758
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.
Amita Malik, Mathematica notebook for generating this sequence and A260793, A291275-A291284
Amita Malik, List of all primes up to 10000 in this sequence and in A260793, A291275-A291284, together with Mathematica code.
E. Rowland, R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013.

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

NeverDividesLucasSeqQ[a_, p_] := And @@ Table[Mod[a[n], p]>0, {n, 0, p-1}];
A3[a_, b_, c_, n_ /; n < 0] = 0;
A3[a_, b_, c_, 0] = 1;
A3[a_, b_, c_, n_] := A3[a, b, c, n] = (((2n - 1)(a (n-1)^2 + a (n-1) + b)) A3[a, b, c, n-1] - c (n-1)^3 A3[a, b, c, n-2])/n^3;
A3[a_, b_, c_, d_, n_ /; n < 0] = 0;
Agamma[n_] := A3[17, 5, 1, n];
Select[Range[1000], PrimeQ[#] && NeverDividesLucasSeqQ[Agamma, #]&] (* Jean-François Alcover, Aug 05 2018, copied from Amita Malik's notebook *)

Terms a(16) onwards computed by Amita Malik - N. J. A. Sloane, Aug 21 2017

A291275 Primes p such that p does not divide any term of the Apéry-like sequence A005258.

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 41, 61, 73, 89, 101, 109, 137, 149, 173, 181, 197, 229, 233, 269, 277, 313, 337, 349, 353, 373, 397, 401, 409, 433, 457, 461, 541, 557, 601, 613, 641, 661, 673, 677, 701, 709, 733, 761, 769, 797, 821, 829, 853, 857, 877, 929, 941, 977

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5.

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

maxPrime = 977;
maxPi = PrimePi @ maxPrime;
okQ[p_] := AllTrue[Range[3 maxPi (* coeff 3 is empirical *)], GCD[HypergeometricPFQ[{# + 1, -#, -#}, {1, 1}, 1], p] == 1&];
Select[Prime[Range[maxPi]], okQ] (* Jean-François Alcover, Jan 13 2020 *)

A291284 Primes p such that p does not divide any term of the Apery-like sequence A290576.

Original entry on oeis.org

2, 5, 7, 13, 17, 19, 29, 37, 43, 47, 59, 61, 67, 71, 83, 89, 101, 109, 127, 139, 149, 167, 173, 191, 211, 233, 239, 241, 251, 257, 271, 277, 281, 307, 311, 313, 331, 337, 347, 349, 353, 359, 373, 379, 383, 409, 419, 421, 431, 433, 443

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A260791 Type II primes (conjectured).

Original entry on oeis.org

5, 7, 13, 23, 31, 37, 47, 53, 71, 101, 103, 167, 181, 191, 197, 199, 223, 229, 263, 271, 293, 317, 349, 383, 397, 431, 439, 461, 479, 503, 509, 541, 557, 599, 607, 613, 647, 653, 677, 709, 719, 727, 733, 743, 751, 757, 797, 821, 823, 839, 877, 887, 911, 919, 991, 997

Offset: 1

N. J. A. Sloane, Aug 05 2015

nonn

See Schulte et al. (2014) for the precise definition.

A. Schulte, S. VanSchalkwyk, A. Yang, On the divisibility and valuations of the Franel numbers, in MSRI-UP Research Reports, 2014.
A. Schulte, S. VanSchalkwyk, A. Yang, On the divisibility and valuations of the Franel numbers, Examples of Outstanding Student Posters, MAA.

Cf. A260793, A260792.

A260792 Type III primes.

Original entry on oeis.org

2, 29, 41, 59, 61, 67, 73, 79, 107, 109, 127, 131, 149, 151, 157, 173, 179, 227, 239, 251, 257, 269, 277, 281, 311, 337, 359, 367, 373, 389, 421, 449, 457, 463, 467, 487, 521, 563, 571, 577, 593, 631, 659, 661, 683, 691, 701, 739, 761, 769, 773, 787, 809, 829, 853, 857, 863, 883, 907, 937, 941, 967, 971, 983

Offset: 1

N. J. A. Sloane, Aug 05 2015

nonn

All primes that are neither in A260791 (Type II) nor in A260793 (type I).

A. Schulte, S. VanSchalkwyk, A. Yang, On the divisibility and valuations of the Franel numbers, in MSRI-UP Research Reports, 2014, p 127.
A. Schulte, S. VanSchalkwyk, A. Yang, On the divisibility and valuations of the Franel numbers, Examples of Outstanding Student Posters, MAA.

Cf. A260793, A260791, A000172.

A291276 Primes p such that p does not divide any term of the Apery-like sequence A002893.

Original entry on oeis.org

2, 7, 13, 37, 61, 73, 109, 127, 157, 163, 193, 211, 223, 229, 271, 283, 307, 313, 331, 337, 349, 367, 379, 409, 421, 433, 463, 487, 499, 523, 577, 607, 613, 619, 631, 661, 673, 691, 727, 733, 751, 757, 769, 787, 823, 829, 853, 883, 907, 919, 1021, 1039

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291277 Primes p such that p does not divide any term of the Apery-like sequence A081085.

Original entry on oeis.org

3, 11, 17, 19, 43, 59, 73, 83, 89, 107, 179, 211, 227, 233, 241, 257, 307, 331, 337, 379, 401, 409, 419, 433, 449, 457, 467, 521, 547, 563, 577, 587, 593, 601, 619, 641, 643, 683, 691, 739, 761, 769, 811, 827, 859, 881, 883, 929, 937, 947, 953

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

A291278 Primes p such that p does not divide any term of the Apery-like sequence A006077.

Original entry on oeis.org

2, 5, 13, 17, 29, 37, 41, 61, 73, 97, 101, 113, 137, 149, 157, 173, 181, 197, 229, 241, 257, 277, 313, 317, 349, 353, 389, 409, 421, 433, 449, 457, 461, 509, 541, 569, 577, 593, 613, 641, 653, 661, 673, 709, 757, 761, 769, 797, 809, 829, 853, 857

Offset: 1

N. J. A. Sloane, Aug 21 2017

nonn

Amita Malik and Armin Straub, Mathematica notebook for generating A133370 and A260793, A291275-A291284
Amita Malik and Armin Straub, Lists of all primes up to 10000 in A133370 and A260793, A291275-A291284, together with Mathematica code.
Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5

For primes that do not divide the terms of the sequences A000172, A005258, A002893, A081085, A006077, A093388, A125143, A229111, A002895, A290575, A290576, A005259 see A260793, A291275-A291284 and A133370 respectively.

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A290575 Apéry-like numbers Sum_{k=0..n} (C(n,k) * C(2*k,n))^2.

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A290576 Apéry-like numbers Sum_{k=0..n} Sum_{l=0..n} (C(n,k)^2*C(n,l)*C(k,l)*C(k+l,n)).

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A133370 Primes p such that p does not divide any term of the Apery sequence A005259 .

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A291275 Primes p such that p does not divide any term of the Apéry-like sequence A005258.

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A291284 Primes p such that p does not divide any term of the Apery-like sequence A290576.

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A260791 Type II primes (conjectured).

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A260792 Type III primes.

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A291276 Primes p such that p does not divide any term of the Apery-like sequence A002893.

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A291277 Primes p such that p does not divide any term of the Apery-like sequence A081085.

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A291278 Primes p such that p does not divide any term of the Apery-like sequence A006077.

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A290576 Apéry-like numbers Sum_{k=0..n} Sum_{l=0..n} (C(n,k)^2C(n,l)C(k,l)*C(k+l,n)).