This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A244973 #48 Mar 17 2025 03:32:47
%S A244973 1,-1,-1,17,-65,49,881,-5489,12223,42785,-479951,1746271,440881,
%T A244973 -39651457,212039855,-326783183,-2817155137,23175692033,-68726927071,
%U A244973 -128775914225,2285692892785,-10156877725985,6169206210815,196882990135745,-1274770281690575
%N A244973 a(n) = Sum_{k=0..n} (-1)^k*C(n, k)^2*C(2*k, k), where C(n, k) denotes the binomial coefficient n!/(k!*(n-k)!).
%C A244973 Zhi-Wei Sun introduced this sequence in arXiv:1407.0967. For any prime p > 5, he proved that Sum_{k=1..p-1} a(k)/k^2 == 0 (mod p) and Sum_{k=1..p-1} a(k)/k == 0 (mod p^2). This is quite similar to Wolstenholme's congruences Sum_{k=1..p-1} 1/k^2 == 0 (mod p) and Sum_{k=1..p-1} 1/k == 0 (mod p^2) for any prime p > 3.
%C A244973 Conjecture: For any prime p > 5 and positive integer n, the number (a(p*n) - a(n))/(p*n)^3 is always a p-adic integer.
%C A244973 The author proved a weaker version of this in arXiv:1610.03384. - _Zhi-Wei Sun_, Nov 12 2016
%H A244973 Zhi-Wei Sun, <a href="/A244973/b244973.txt">Table of n, a(n) for n = 0..200</a>
%H A244973 V. J. Guo, G.-S. Mao and H. Pan, <a href="http://dx.doi.org/10.1080/10236198.2016.1188088">Proof of a conjecture involving Sun polynomials</a>, J. Difference Equ. Appl., 22(2016), no. 8, 1184-1197; also <a href="https://arxiv.org/abs/1511.04005">arXiv:1511.04005 [math.NT]</a>, 2015.
%H A244973 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1407.0967">Congruences involving g_n(x) = sum_{k=0}^n C(n,k)^2*C(2k,k)*x^k</a>, arXiv:1407.0967 [math.NT], 2014-2016.
%H A244973 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1007/s11139-015-9727-3">Congruences involving g_n(x) = sum_{k=0}^n C(n,k)^2*C(2k,k)*x^k</a>, Ramanujan J. 40 (2016), no. 3, 511-533.
%H A244973 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1610.03384">Supercongruences involving Lucas sequences</a>, arXiv:1016.03384 [math.NT], 2016.
%F A244973 Recurrence (obtained via the Zeilberger algorithm):
%F A244973 (n+3)^2*(4n+5)*a(n+3) + (20n^3+125n^2+254n+165)*a(n+2) + (76n^3+399n^2+678n+375)*a(n+1) - 25*(n+1)^2*(4n+9)*a(n) = 0.
%F A244973 Lim_sup_{n->oo} |a(n)|^(1/n) = 5. - _Vaclav Kotesovec_, Jul 13 2014
%F A244973 a(n) = Sum_{k = 0..n} (-1)^(n-k)*C(n,2*k)^2*C(2*k,k) = Sum_{k = 0..n} (-1)^(n-k)*C(n,k)*C(n,2*k)*C(n-k,k). - _Zhi-Wei Sun_, Nov 12 2016
%F A244973 Conjecture: a(n) = Sum_{k = 0..n} binomial(n, k)*b(k), where b(n) = Sum_{k = 0..n} (-1)^k*binomial(n, k)^2*binomial(2*k, n). [Added Mar 16 2025: this conjecture can be verified using the MulZeil procedure in Doron Zeilberger's MultiZeilberger package for Maple]. - _Peter Bala_, Jul 19 2024
%e A244973 a(3) = 17 since C(3,0)^2*C(2*0,0) - C(3,1)^2*C(2,1) + C(3,2)^2*C(4,2) - C(3,3)^2*C(6,3) = 1 - 18 + 54 - 20 = 17.
%p A244973 a := n -> hypergeom([1/2, -n, -n], [1, 1], -4):
%p A244973 seq(simplify(a(n)), n = 0..24); # _Peter Luschny_, Mar 16 2025
%t A244973 s[n_]:=Sum[Binomial[n,k]^2*Binomial[2k,k](-1)^k,{k,0,n}]
%t A244973 Table[s[n],{n,0,20}]
%o A244973 (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n,k)^2*binomial(2*k,k)); \\ _Michel Marcus_, Nov 13 2016
%Y A244973 Cf. A000172, A002893, A275027, A277640.
%K A244973 sign,easy
%O A244973 0,4
%A A244973 _Zhi-Wei Sun_, Jul 08 2014