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A260667 a(n) = (Sum_{k=0..n-1}(2k+1)*S(k,n)^2)/n^2, where S(k,x) denotes the polynomial Sum_{j=0..k}binomial(k,j)*binomial(x,j)*binomial(x+j,j). 34
1, 37, 1737, 102501, 6979833, 523680739, 42129659113, 3572184623653, 315561396741609, 28807571694394593, 2701627814373536601, 259121323945378645947, 25330657454041707496017, 2516984276442279642274311, 253667099464270541534450025, 25884030861250181046253181349, 2670255662315910532447096232073 (list; graph; refs; listen; history; text; internal format)



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 (Sum_{k=0..n-1}(2k+1)*S(k,x)^2)/n^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


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}binom(n,k)^2*binom(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.


a(2) = 37 since (Sum_{k=0,1}(2k+1)*S(k,2)^2)/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 + ...


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}]


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.)

Sequence in context: A009695 A201956 A322097 * A130013 A088872 A025762

Adjacent sequences:  A260664 A260665 A260666 * A260668 A260669 A260670




Zhi-Wei Sun, Nov 14 2015



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Last modified February 28 14:02 EST 2021. Contains 341707 sequences. (Running on oeis4.)