A002893 -id:A002893 - cp's OEIS frontend

A291280 Primes p such that p does not divide any term of the Apery-like sequence A125143.

2, 5, 7, 11, 13, 19, 29, 41, 47, 61, 67, 71, 73, 89, 97, 101, 103, 131, 137, 139, 149, 157, 163, 167, 173, 179, 181, 193, 197, 199, 211, 223, 227, 229, 233, 241, 251, 257, 263, 269, 281, 283, 293, 317, 331, 347, 349, 359, 367, 373, 379, 383

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.

A291281 Primes p such that p does not divide any term of the Apery-like sequence A229111.

Original entry on oeis.org

2, 3, 17, 19, 23, 31, 47, 53, 61, 107, 109, 113, 137, 139, 151, 173, 197, 199, 211, 227, 229, 233, 241, 257, 263, 293, 317, 347, 353, 383, 421, 439, 443, 467, 499, 541, 587, 593, 619, 647, 661, 677, 683, 691, 751, 769, 773, 857, 919

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.

A291282 Primes p such that p does not divide any term of the Apery-like sequence A002895.

Original entry on oeis.org

3, 5, 13, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 83, 89, 101, 103, 107, 109, 127, 131, 137, 163, 167, 173, 181, 191, 193, 199, 229, 233, 239, 241, 251, 257, 269, 307, 311, 317, 337, 347, 349, 359, 367, 373, 409, 419, 421, 449, 457

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.

A291283 Primes p such that p does not divide any term of the Apery-like sequence A290575.

Original entry on oeis.org

3, 7, 13, 19, 23, 29, 31, 37, 43, 47, 61, 67, 73, 83, 89, 101, 103, 107, 109, 113, 127, 131, 139, 149, 157, 167, 179, 191, 193, 197, 223, 227, 229, 241, 251, 257, 271, 281, 293, 307, 311, 313, 353, 367, 373, 379, 389, 401, 419, 431, 433

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.

A327994 a(n) is the constant term in the expansion of (1 + (1+x)(1+y) + (1+1/x)(1+1/y) + (1+1/x^2)*(1+1/y^2))^n.

Original entry on oeis.org

1, 4, 22, 157, 1342, 12694, 126601, 1301353, 13647622, 145280776, 1564974682, 17022150688, 186647326849, 2060538384454, 22880172406939, 255336518921572, 2861956399934038, 32201882791307904, 363561893114484868, 4117138703073839926, 46751784944876067562

Offset: 0

Peter Luschny, Oct 28 2019

nonn

Cf. A002893.

F[n_] := Expand[(1+(1+x)(1+y) + (1+1/x)(1+1/y) + (1+1/x^2)(1+1/y^2))^n];
a[n_] := SeriesCoefficient[F[n], {x, 0, 0}, {y, 0, 0}]; a /@ Range[0, 20]

A338934 Square array T(i,j) = Sum_{k=0...min(i,j)} C(i,k)C(j,k)C(2*k,k) (i>=0,j>=0), read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 7, 15, 7, 1, 1, 9, 31, 31, 9, 1, 1, 11, 53, 93, 53, 11, 1, 1, 13, 81, 213, 213, 81, 13, 1, 1, 15, 115, 411, 639, 411, 115, 15, 1, 1, 17, 155, 707, 1551, 1551, 707, 155, 17, 1, 1, 19, 201, 1121, 3239, 4653, 3239, 1121, 201, 19, 1

Offset: 0

Ludovic Schwob, Nov 16 2020

T(i,j)*C(i+j,i) is the number of ways to write the vector (i,i,j,j) as a sum of vectors containing two occurrences of the number 1.

Up to order, the number of different sums is A106255(i+1,j+1).

			There are T(1,1)*C(2,1)=6 ways to write the vector (1,1,1,1) as a sum of vectors containing two occurrences of the number 1 : (1,1,0,0)+(0,0,1,1), (0,0,1,1)+(1,1,0,0), (1,0,1,0)+(0,1,0,1), (0,1,0,1)+(1,0,1,0), (1,0,0,1)+(0,1,1,0), (0,1,1,0)+(1,0,0,1).
The square array T(i,j) (i >= 0, j >= 0) begins:
  1,  1,  1,   1,    1,    1, ...
  1,  3,  5,   7,    9,   11, ...
  1,  5, 15,  31,   53,   81, ...
  1,  7, 31,  93,  213,  411, ...
  1,  9, 53, 213,  639, 1551, ...
  1, 11, 81, 411, 1551, 4653, ...
  ...

Central diagonal terms give A002893.

Antidiagonal sums give A097893.

T[i_,j_]:=Sum[Binomial[i,k]Binomial[j,k]Binomial[2k,k],{k,0,Min[i,j]}]; Flatten[Table[T[i-j,j],{i,0,10},{j,0,i}]] (* Stefano Spezia, Nov 17 2020 *)

More terms from Stefano Spezia, Nov 17 2020

A349769 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(k,floor(k/2)).

Original entry on oeis.org

1, 2, 7, 31, 143, 686, 3417, 17382, 89791, 470134, 2487593, 13273921, 71341921, 385786298, 2097096263, 11451611919, 62784274623, 345436875758, 1906568766489, 10552637998329, 58556298508449, 325676578717698, 1815140080977303, 10135993961893674

Offset: 0

Ilya Gutkovskiy, Nov 29 2021

Cf. A001405, A002893, A005773, A026375.

Table[Sum[Binomial[n, k]^2 Binomial[k, Floor[k/2]], {k, 0, n}], {n, 0, 22}]

a(n) = sum(k=0, n, binomial(n,k)^2 * binomial(k,k\2)); \\ Michel Marcus, Nov 29 2021

From Vaclav Kotesovec, Nov 29 2021: (Start)
D-finite recurrence: n*(n+1)^2*(32*n^4 - 208*n^3 + 456*n^2 - 380*n + 81)*a(n) = 2*n*(64*n^6 - 320*n^5 + 344*n^4 + 288*n^3 - 332*n^2 - 363*n + 243)*a(n-1) + 2*(n-1)*(32*n^6 - 80*n^5 - 624*n^4 + 2492*n^3 - 2257*n^2 - 596*n + 927)*a(n-2) + 2*(n-2)*(832*n^6 - 6272*n^5 + 16360*n^4 - 15456*n^3 - 1016*n^2 + 8375*n - 3195)*a(n-3) - 9*(n-3)^2*(n-2)*(32*n^4 - 80*n^3 + 24*n^2 + 36*n - 19)*a(n-4).
a(n) ~ (1 + sqrt(2))^(2*n + 3/2) / (2*Pi*n). (End)

A381199 a(n) = (4n)!/((n!)^2(2n)!)Sum_{k=0..n} binomial(n,k)^2binomial(2k,k).

Original entry on oeis.org

1, 36, 6300, 1718640, 575675100, 216636756336, 87874675224336, 37563969509352000, 16692217815436148700, 7642084994921759382000, 3582530520581922083974800, 1712083670316898167464884800, 831357643152788660610464490000, 409154554816583487288034143528000, 203690783136217174743485058666840000

Offset: 0

Stefano Spezia, Feb 16 2025

nonn

S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 24.

Cf. A000897, A000984, A001044, A002893, A007318, A008459, A010050, A100733.

a[n_]:=(4n)!/((n!)^2*(2n)!)*Sum[Binomial[n,k]^2Binomial[2k,k],{k,0,n}]; Array[a,15,0]

a(n) = (4*n)!*hypergeom([1/2, -n, -n], [1, 1], 4)/((n!)^2*(2*n)!).
D-finite with recurrence n^4*a(n) -4*(4*n-1)*(4*n-3)*(10*n^2-10*n+3)*a(n-1) +144*(4*n-5)*(4*n-3)*(4*n-7)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Feb 18 2025
a(n) ~ 2^(6*n - 1/2) * 3^(2*n + 3/2) / (4*Pi^2*n^2). - Vaclav Kotesovec, May 29 2025

cp's OEIS Frontend

A291280 Primes p such that p does not divide any term of the Apery-like sequence A125143.

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A291281 Primes p such that p does not divide any term of the Apery-like sequence A229111.

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A291282 Primes p such that p does not divide any term of the Apery-like sequence A002895.

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A291283 Primes p such that p does not divide any term of the Apery-like sequence A290575.

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A327994 a(n) is the constant term in the expansion of (1 + (1+x)*(1+y) + (1+1/x)*(1+1/y) + (1+1/x^2)*(1+1/y^2))^n.

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A338934 Square array T(i,j) = Sum_{k=0...min(i,j)} C(i,k)*C(j,k)*C(2*k,k) (i>=0,j>=0), read by antidiagonals.

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A349769 a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(k,floor(k/2)).

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A381199 a(n) = (4*n)!/((n!)^2*(2*n)!)*Sum_{k=0..n} binomial(n,k)^2*binomial(2*k,k).

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A327994 a(n) is the constant term in the expansion of (1 + (1+x)(1+y) + (1+1/x)(1+1/y) + (1+1/x^2)*(1+1/y^2))^n.

A338934 Square array T(i,j) = Sum_{k=0...min(i,j)} C(i,k)C(j,k)C(2*k,k) (i>=0,j>=0), read by antidiagonals.

A381199 a(n) = (4n)!/((n!)^2(2n)!)Sum_{k=0..n} binomial(n,k)^2binomial(2k,k).