A125143 -id:A125143 - cp's OEIS frontend

A143415 Another sequence of Apery-like numbers for the constant 1/e: a(n) = 1/(n+1)!Sum_{k = 0..n-1} C(n-1,k)(2*n-k)!.

0, 1, 5, 41, 481, 7421, 142601, 3288205, 88577021, 2731868921, 94969529101, 3675200329841, 156725471006105, 7302990263511541, 369216917569411601, 20130327811188977621, 1177435382675193700021, 73546210385434763486705

Offset: 0

Peter Bala, Aug 14 2008

This sequence is a modified version of A143414.

Seiichi Manyama, Table of n, a(n) for n = 0..366

Cf. A143413, A143414.

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.)

a := n -> 1/(n+1)!*add (binomial(n-1,k)*(2*n-k)!,k = 0..n-1): seq(a(n),n = 0..19);
# Alternative:
A143415 := n -> `if`(n=0, 0, ((2*n)!/(n+1)!)*hypergeom([1-n], [-2*n], 1)):
seq(simplify(A143415(n)), n = 0..17); # Peter Luschny, May 14 2020

Table[(1/(n+1)!)*Sum[Binomial[n-1,k]*(2*n-k)!, {k,0,n-1}], {n,0,50}] (* G. C. Greubel, Oct 24 2017 *)

for(n=0,25, print1((1/(n+1)!)*sum(k=0,n-1, binomial(n-1,k)*(2*n-k)!), ", ")) \\ G. C. Greubel, Oct 24 2017

a(n) = 1/(n+1)!*sum {k = 0..n-1} C(n-1,k)*(2*n-k)!.
a(n) = 1/(n*(n+1))*A143414(n) for n > 0.
Recurrence relation: a(0) = 0, a(1) = 1, (n-1)*(n+1)*a(n) - (n-2)*n*a(n-2) = (2*n-1)*(2*n^2-2*n+1)*a(n-1) for n >= 2. 1/e = 1/2 - 2 * Sum_{n = 1..inf} (-1)^(n+1)/(n*(n+2)*a(n)*a(n+1)) = 1/2 - 2*[1/(3*1*5) - 1/(8*5*41) + 1/(15*41*481) - 1/(24*481*7421) + ...] .
Conjectural congruences: for r >= 0 and prime p, calculation suggests the congruences a(p^r*(p+1)) == a(p^r) (mod p^(r+1)) may hold.
a(n) = ((2*n)!/(n+1)!)*hypergeom([1-n], [-2*n], 1) for n > 0. - Peter Luschny, May 14 2020

A219692 a(n) = Sum_{j=0..floor(n/3)} (-1)^j C(n,j) * C(2j,j) * C(2n-2j,n-j) * (C(2n-3j-1,n) + C(2n-3j,n)).

Original entry on oeis.org

2, 6, 54, 564, 6390, 76356, 948276, 12132504, 158984694, 2124923460, 28877309604, 398046897144, 5554209125556, 78328566695736, 1114923122685720, 15999482238880464, 231253045986317814, 3363838379489630916

Offset: 0

Jason Kimberley, Nov 25 2012

This sequence is s_18 in Cooper's paper.
This is one of the Apery-like sequences - see Cross-references. - Hugo Pfoertner, Aug 06 2017
Every prime eventually divides some term of this sequence. - Amita Malik, Aug 20 2017

G. C. Greubel, Table of n, a(n) for n = 0..830 (terms 0..254 from Jason Kimberley)
S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).
Ofir Gorodetsky, New representations for all sporadic Apéry-like sequences, with applications to congruences, arXiv:2102.11839 [math.NT], 2021. See s18 p. 3.
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.)

Magma
```
s_18 := func where C is Binomial;
```

Table[Sum[(-1)^j*Binomial[n,j]*Binomial[2j,j]*Binomial[2n-2j, n-j]* (Binomial[2n-3j-1,n] +Binomial[2n-3j,n]), {j,0,Floor[n/3]}], {n,0,20}] (* G. C. Greubel, Oct 24 2017 *)

{a(n) = sum(j=0,floor(n/3), (-1)^j*binomial(n,j)*binomial(2*j,j)* binomial(2*n-2*j,n-j)*(binomial(2*n-3*j-1,n) +binomial(2*n-3*j,n)))}; \\ G. C. Greubel, Apr 02 2019

[sum((-1)^j*binomial(n,j)*binomial(2*j,j)*binomial(2*n-2*j,n-j)* (binomial(2*n-3*j-1,n)+binomial(2*n-3*j,n)) for j in (0..floor(n/3))) for n in (0..20)] # G. C. Greubel, Apr 02 2019

1/Pi
= 2*3^(-5/2) Sum {k>=0} (n a(n)/18^n) [Cooper, equation (42)]
= 2*3^(-5/2) Sum {k>=0} (n a(n)/A001027(n)).
G.f.: 1+hypergeom([1/8, 3/8],[1],256*x^3/(1-12*x)^2)^2/sqrt(1-12*x). - Mark van Hoeij, May 07 2013
Conjecture D-finite with recurrence: n^3*a(n) -2*(2*n-1)*(7*n^2-7*n+3)*a(n-1) +12*(4*n-5)*(n-1)* (4*n-3)*a(n-2)=0. - R. J. Mathar, Jun 14 2016
a(n) ~ 3 * 2^(4*n + 1/2) / (Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Mar 08 2023

A260667 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)S(k,n)^2, where S(k,x) denotes the polynomial Sum_{j=0..k} binomial(k,j)binomial(x,j)*binomial(x+j,j).

Original entry on oeis.org

1, 37, 1737, 102501, 6979833, 523680739, 42129659113, 3572184623653, 315561396741609, 28807571694394593, 2701627814373536601, 259121323945378645947, 25330657454041707496017, 2516984276442279642274311, 253667099464270541534450025, 25884030861250181046253181349, 2670255662315910532447096232073

Offset: 1

Zhi-Wei Sun, Nov 14 2015

nonn

Conjecture: For k = 0,1,2,... define S(k,x):= Sum_{j=0..k} binomial(k,j)*binomial(x,j)*binomial(x+j,j).
(i) For any integer n > 0, the polynomial (1/n^2) * Sum_{k=0..n-1}(2k+1)*S(k,x)^2 is integer-valued (and hence a(n) is always integral).
(ii) Let r be 0 or 1, and let x be any integer. Then, for any positive integers m and n, we have the congruence
   Sum_{k=0..n-1} (-1)^(k*r)*(2k+1)*S(k,x)^(2m) == 0 (mod n).
(iii) For any odd prime p, we have Sum_{k=0..p-1} S(k,-1/2)^2 == (-1/p)(1-7*p^3*B_{p-3}) (mod p^4), where (a/p) is the Legendre symbol, and B_0,B_1,B_2,... are Bernoulli numbers. Also, for any prime p > 3 we have Sum_{k=0..p-1} S(k,-1/3)^2 == p - (14/3)*(p/3)*p^3*B_{p-2}(1/3) (mod p^4), where B_n(x) denotes the Bernoulli polynomial of degree n; Sum_{k=0..p-1} S(k,-1/4)^2 == (2/p)*p - 26*(-2/p)*p^3*E_{p-3} (mod p^4), where E_0,E_1,E_2,... are Euler numbers; Sum_{k=0..p-1} S(k,-1/6)^2 == (3/p)*p - (155/12)*(-1/p)*p^3*B_{p-2}(1/3) (mod p^4).
Our conjecture is motivated by a conjecture of Kimoto and Wakayama which states that Sum_{k=0..p-1} S(k,-1/2)^2 == (-1/p) (mod p^3) for any odd prime p. The Kimoto-Wakayama conjecture was confirmed by Long, Osburn and Swisher in 2014.
For more related conjectures, see Sun's paper arXiv.1512.00712. - Zhi-Wei Sun, Dec 03 2015

			a(2) = 37 since (1/2^2) * Sum_{k=0..1} (2k+1)*S(k,2)^2 = (S(0,2)^2 + 3*S(1,2)^2)/4 = (1^2 + 3*7^2)/4 = 148/4 = 37.
G.f. = x + 37*x^2 + 1737*x^3 + 102501*x^4 + 6979833*x^5 + 523680739*x^6 + ...

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
K. Kimoto and M. Wakayama, Apéry-like numbers arising from special values of spectral zeta function for non-commutative harmonic oscillators, Kyushu J. Math. 60(2006), no.2, 383-404. (Cf. the formula (6.19).)
L. Long, R. Osburn and H. Swisher, On a conjecture of Kimoto and Wakayama, arXiv:1404.4723 [math.NT], 2014.
Z.-W. Sun, Supercongruences involving products of two binomial coefficients, arXiv:1011.6676 [math.NT], 2010-2013; Finite Fields Appl. 22(2013), 24-44.
Z.-W. Sun, Congruences involving g_n(x)=sum_{k=0..n} binomial(n,k)^2*binomial(2k,k)*x^k, Ramanujan J., in press. Doi: 10.1007/s11139-015-9727-3.
Z.-W. Sun, Supercongruences involving dual sequences, arXiv:1502.00712 [math.NT], 2015.

Cf. A000290, A027641, A027642, A122045.

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.)

# Implementing Mark van Hoeij's formula.
c := n -> binomial(2*n, n)/(n + 1):
h := n -> simplify(hypergeom([-n,-n,-n], [1,-2*n], 1)):
b := n -> c(n)^2*((n+11)*(2+4*n)^2*h(n+1)^2-2*(n+1)*(11*n+16)*(1+2*n)*h(n)*h(n+1)-h(n)^2*(n+1)^3)/(25*(n+2)):
a := n -> b(n-1): seq(a(n), n = 1..17);  # Peter Luschny, Nov 11 2022

S[k_,x_]:=S[k,x]=Sum[Binomial[k,j]Binomial[x,j]Binomial[x+j,j],{j,0,k}]
a[n_]:=a[n]=Sum[(2k+1)*S[k,n]^2,{k,0,n-1}]/n^2
Do[Print[n," ",a[n]],{n,1,17}]

a(n) ~ phi^(10*n + 3) / (10 * Pi^2 * n^3), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 06 2021
Conjecture: a(p-1) == 1 (mod p^3) for all primes p >= 5. - Peter Bala, Aug 15 2022
a(n) = ((n+10)*A005258(n)^2 - (11*n+5)*A005258(n)*A005258(n-1) - n*A005258(n-1)^2)/(25*(n+1)). - Mark van Hoeij, Nov 11 2022

A262177 Decimal expansion of Q_5 = zeta(5) / (Sum_{k>=1} (-1)^(k+1) / (k^5 * binomial(2k, k))), a conjecturally irrational constant defined by an Apéry-like formula.

Original entry on oeis.org

2, 0, 9, 4, 8, 6, 8, 6, 2, 2, 0, 1, 0, 0, 3, 6, 9, 9, 3, 8, 5, 0, 2, 4, 9, 2, 9, 3, 7, 3, 2, 9, 4, 1, 6, 3, 0, 2, 9, 6, 7, 5, 8, 7, 4, 8, 5, 6, 7, 7, 8, 1, 8, 2, 7, 4, 0, 1, 2, 7, 5, 8, 7, 8, 3, 7, 4, 3, 8, 0, 0, 7, 8, 7, 6, 8, 4, 6, 8, 1, 5, 6, 3, 2, 0, 6, 0, 4, 4, 2, 3, 2, 0, 9, 0, 4, 3, 1, 3, 6, 9, 3, 1

Offset: 1

Jean-François Alcover, Sep 14 2015

The similar constant Q_3 = zeta(3) / (Sum_{k>=1} (-1)^(k+1) / (k^3 * binomial(2k, k))) evaluates to 5/2.

			2.09486862201003699385024929373294163029675874856778182740127587837438...

G. C. Greubel, Table of n, a(n) for n = 1..5000
David Bailey, Jonathan Borwein, David Bradley, Experimental determination of Apéry-like identities for zeta(2n+2), arXiv:math/0505270 [math.NT], 2005.

Cf. A013663.

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.)

Q5 = Zeta[5]/Sum[(-1)^(k+1)/(k^5*Binomial[2k, k]), {k, 1, Infinity}]; RealDigits[Q5, 10, 103] // First

zeta(5)/suminf(k=1, (-1)^(k+1)/(k^5*binomial(2*k,k))) \\ Michel Marcus, Sep 14 2015

Equals 2*zeta(5)/6F5(1,1,1,1,1,1; 3/2,2,2,2,2; -1/4).

A260793 Primes p such that p does not divide any term of the Apéry-like sequence A000172 (also known as Type I primes).

Original entry on oeis.org

3, 11, 17, 19, 43, 83, 89, 97, 113, 137, 139, 163, 193, 211, 233, 241, 283, 307, 313, 331, 347, 353, 379, 401, 409, 419, 433, 443, 491, 499, 523, 547, 569, 587, 601, 617, 619, 641, 643, 673, 811, 827, 859, 881, 929, 947, 953, 977, 1009, 1019, 1033, 1049, 1051

Offset: 1

N. J. A. Sloane, Aug 05 2015

nonn

See Schulte et al. (2014) for the precise definition of Type I primes.

Amita Malik and Armin Straub, Divisibility properties of sporadic Apéry-like numbers, Research in Number Theory, 2016, 2:5
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
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. A260791, A260792.

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

maxPrime = 1051;
maxPi = PrimePi @ maxPrime;
okQ[p_] := AllTrue[Range[3 maxPi (* coeff 3 is empirical *)], GCD[HypergeometricPFQ[{-#, -#, -#}, {1, 1}, -1], p] == 1&];
Select[Prime[Range[maxPi]], okQ] (* Jean-François Alcover, Jan 13 2020 *)

Edited by N. J. A. Sloane, Aug 22 2017

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.

A318105 Triangle read by rows: T(n,k) = (4n - 3k)!/((n-k)!^4*k!).

Original entry on oeis.org

1, 24, 1, 2520, 120, 1, 369600, 22680, 360, 1, 63063000, 4804800, 113400, 840, 1, 11732745024, 1072071000, 33633600, 415800, 1680, 1, 2308743493056, 246387645504, 9648639000, 168168000, 1247400, 3024, 1, 472518347558400, 57718587326400, 2710264100544, 61108047000, 672672000, 3243240, 5040, 1

Offset: 0

Gheorghe Coserea, Oct 15 2018

Diagonal of rational function R(x,y,z,w,t) = 1/(1 - (x+y+z+w + t*x*y*z*w)) with respect to x,y,z,w, i.e., T(n,k) = [(xyzw)^n*t^k] R(x,y,z,w,t).

Annihilating differential operator: x^2*(3*t*x + 1)^2*((t*x - 1)^4 - 256*x)*Dx^3 + 3*x*(3*t*x + 1)*((t*x - 1)^3*(6*t^2*x^2 + 3*t*x - 1) - 384*x*(t*x + 1))*Dx^2 + (t*x - 1)*((t*x - 1)*(63*t^4*x^4 + 66*t^3*x^3 - 18*t*x + 1) + 48*x*(15*t*x + 17))*Dx + (t*x - 1)*(t*(9*t^4*x^4 + 12*t^3*x^3 + 6*t^2*x^2 - 12*t*x + 1) - 24*(15*t*x - 1)).

			A(x;t) = 1 + (24 + t)*x + (2520 + 120*t + t^2)*x^2 + (369600 + 22680*t + 360*t^2 + t^3)*x^3 + ...
Triangle starts:
n\k [0]            [1]           [2]         [3]        [4]      [5]   [6]
[0] 1;
[1] 24,            1;
[2] 2520,          120,          1;
[3] 369600,        22680,        360,        1;
[4] 63063000,      4804800,      113400,     840,       1;
[5] 11732745024,   1072071000,   33633600,   415800,    1680,    1;
[6] 2308743493056, 246387645504, 9648639000, 168168000, 1247400, 3024, 1;
[7] ...

Gheorghe Coserea, Rows n=0..100, flattened

Cf. A008977, A082488, A125143, A318107.

t[n_,k_] := (4*n - 3*k)!/((n-k)!^4*k!); Table[t[n, k], {n, 0, 10}, {k , 0,n}] // Flatten  (* Amiram Eldar, Nov 07 2018 *)

T(n, k) = (4*n-3*k)!/(k!*(n-k)!^4);
concat(vector(8, n, vector(n, k, T(n-1, k-1))))
/*
test:
P(n, v='t) = subst(Polrev(vector(n+1, k, T(n, k-1)), 't), 't, v);
diag(expr, N=22, var=variables(expr)) = {
  my(a = vector(N));
  for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
  for (n = 1, N, a[n] = expr;
    for (k = 1, #var, a[n] = polcoef(a[n], n-1)));
  return(a);
};
apply_diffop(p, s) = {
  s=intformal(s);
  sum(n=0, poldegree(p, 'Dx), s=s'; polcoef(p, n, 'Dx) * s);
};
\\ diagonal property:
x='x; y='y; z='z; w='w; t='t;
diag(1/(1 - (x+y+z+w + t*x*y*z*w)), 9, [x, y, z, w]) == vector(9, n, P(n-1))
\\ annihilating diffop:
y = Ser(vector(101, n, P(n-1)), 'x);
p = x^2*(3*t*x + 1)^2*((t*x - 1)^4 - 256*x)*Dx^3 + 3*x*(3*t*x + 1)*((t*x - 1)^3*(6*t^2*x^2 + 3*t*x - 1) - 384*x*(t*x + 1))*Dx^2 + (t*x - 1)*((t*x - 1)*(63*t^4*x^4 + 66*t^3*x^3 - 18*t*x + 1) + 48*x*(15*t*x + 17))*Dx + (t*x - 1)*(t*(9*t^4*x^4 + 12*t^3*x^3 + 6*t^2*x^2 - 12*t*x + 1) - 24*(15*t*x - 1));
0 == apply_diffop(p, y)
*/

Let P_n(t) = Sum_{k=0..n} T(n,k)*t^k. Then A125143(n) = P_n(-27), A008977(n) = P_n(0), A082488(n) = P_n(1).

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.

cp's OEIS Frontend

A143415 Another sequence of Apery-like numbers for the constant 1/e: a(n) = 1/(n+1)!*Sum_{k = 0..n-1} C(n-1,k)*(2*n-k)!.

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A219692 a(n) = Sum_{j=0..floor(n/3)} (-1)^j C(n,j) * C(2j,j) * C(2n-2j,n-j) * (C(2n-3j-1,n) + C(2n-3j,n)).

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A260667 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)*S(k,n)^2, where S(k,x) denotes the polynomial Sum_{j=0..k} binomial(k,j)*binomial(x,j)*binomial(x+j,j).

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A262177 Decimal expansion of Q_5 = zeta(5) / (Sum_{k>=1} (-1)^(k+1) / (k^5 * binomial(2k, k))), a conjecturally irrational constant defined by an Apéry-like formula.

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A260793 Primes p such that p does not divide any term of the Apéry-like sequence A000172 (also known as Type I primes).

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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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A143415 Another sequence of Apery-like numbers for the constant 1/e: a(n) = 1/(n+1)!Sum_{k = 0..n-1} C(n-1,k)(2*n-k)!.

A260667 a(n) = (1/n^2) * Sum_{k=0..n-1} (2k+1)S(k,n)^2, where S(k,x) denotes the polynomial Sum_{j=0..k} binomial(k,j)binomial(x,j)*binomial(x+j,j).

A318105 Triangle read by rows: T(n,k) = (4n - 3k)!/((n-k)!^4*k!).