cp's OEIS Frontend

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).

%I A260667 #58 Mar 28 2025 16:44:57
%S A260667 1,37,1737,102501,6979833,523680739,42129659113,3572184623653,
%T A260667 315561396741609,28807571694394593,2701627814373536601,
%U A260667 259121323945378645947,25330657454041707496017,2516984276442279642274311,253667099464270541534450025,25884030861250181046253181349,2670255662315910532447096232073
%N 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).
%C A260667 Conjecture: For k = 0,1,2,... define S(k,x):= Sum_{j=0..k} binomial(k,j)*binomial(x,j)*binomial(x+j,j).
%C A260667 (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).
%C A260667 (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
%C A260667    Sum_{k=0..n-1} (-1)^(k*r)*(2k+1)*S(k,x)^(2m) == 0 (mod n).
%C A260667 (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).
%C A260667 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.
%C A260667 For more related conjectures, see Sun's paper arXiv.1512.00712. - _Zhi-Wei Sun_, Dec 03 2015
%H A260667 Zhi-Wei Sun, <a href="/A260667/b260667.txt">Table of n, a(n) for n = 1..100</a>
%H A260667 K. Kimoto and M. Wakayama, <a href="http://doi.org/10.2206/kyushujm.60.383">Apéry-like numbers arising from special values of spectral zeta function for non-commutative harmonic oscillators</a>, Kyushu J. Math. 60(2006), no.2, 383-404. (Cf. the formula (6.19).)
%H A260667 L. Long, R. Osburn and H. Swisher, <a href="http://arxiv.org/abs/1404.4723">On a conjecture of Kimoto and Wakayama</a>, arXiv:1404.4723 [math.NT], 2014.
%H A260667 Z.-W. Sun, <a href="https://arxiv.org/abs/1011.6676">Supercongruences involving products of two binomial coefficients</a>, arXiv:1011.6676 [math.NT], 2010-2013; Finite Fields Appl. 22(2013), 24-44.
%H A260667 Z.-W. Sun, <a href="http://dx.doi.org/10.1007/s11139-015-9727-3">Congruences involving g_n(x)=sum_{k=0..n} binomial(n,k)^2*binomial(2k,k)*x^k</a>, Ramanujan J., in press. Doi: 10.1007/s11139-015-9727-3.
%H A260667 Z.-W. Sun, <a href="http://arxiv.org/abs/1512.00712">Supercongruences involving dual sequences</a>, arXiv:1502.00712 [math.NT], 2015.
%F A260667 a(n) ~ phi^(10*n + 3) / (10 * Pi^2 * n^3), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Nov 06 2021
%F A260667 Conjecture: a(p-1) == 1 (mod p^3) for all primes p >= 5. - _Peter Bala_, Aug 15 2022
%F A260667 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
%e A260667 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.
%e A260667 G.f. = x + 37*x^2 + 1737*x^3 + 102501*x^4 + 6979833*x^5 + 523680739*x^6 + ...
%p A260667 # Implementing _Mark van Hoeij_'s formula.
%p A260667 c := n -> binomial(2*n, n)/(n + 1):
%p A260667 h := n -> simplify(hypergeom([-n,-n,-n], [1,-2*n], 1)):
%p A260667 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)):
%p A260667 a := n -> b(n-1): seq(a(n), n = 1..17);  # _Peter Luschny_, Nov 11 2022
%t A260667 S[k_,x_]:=S[k,x]=Sum[Binomial[k,j]Binomial[x,j]Binomial[x+j,j],{j,0,k}]
%t A260667 a[n_]:=a[n]=Sum[(2k+1)*S[k,n]^2,{k,0,n-1}]/n^2
%t A260667 Do[Print[n," ",a[n]],{n,1,17}]
%Y A260667 Cf. A000290, A027641, A027642, A122045.
%Y A260667 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.)
%K A260667 nonn
%O A260667 1,2
%A A260667 _Zhi-Wei Sun_, Nov 14 2015