A244973 a(n) = Sum_{k=0..n} C(n,k)^2*C(2k,k)(-1)^k, where C(n,k) denotes the binomial coefficient n!/(k!(n-k)!).

1, -1, -1, 17, -65, 49, 881, -5489, 12223, 42785, -479951, 1746271, 440881, -39651457, 212039855, -326783183, -2817155137, 23175692033, -68726927071, -128775914225, 2285692892785, -10156877725985, 6169206210815, 196882990135745, -1274770281690575

Offset

0,4

Comments

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.

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.

The author proved a weaker version of this in arXiv:1610.03384. - Zhi-Wei Sun, Nov 12 2016

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..200

V. J. Guo, G.-S. Mao and H. Pan, Proof of a conjecture involving Sun polynomials, J. Difference Equ. Appl., 22(2016), no. 8, 1184-1197; also arXiv:1511.04005 [math.NT], 2015.

Zhi-Wei Sun, Congruences involving g_n(x) = sum_{k=0}^n C(n,k)^2*C(2k,k)*x^k, arXiv:1407.0967 [math.NT], 2014-2016.

Zhi-Wei Sun, Congruences involving  g_n(x) = sum_{k=0}^n C(n,k)^2*C(2k,k)*x^k, Ramanujan J. 40 (2016), no. 3, 511-533.

Zhi-Wei Sun, Supercongruences involving Lucas sequences, arXiv:1016.03384 [math.NT], 2016.

Formula

Recurrence (obtained via the Zeilberger algorithm):

(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.

Lim sup n->infinity |a(n)|^(1/n) = 5. - Vaclav Kotesovec, Jul 13 2014

a(n) = Sum_{k=0..n}C(n,2k)^2*C(2k,k)*(-1)^(n-k) = Sum_{k=0..n}C(n,k)*C(n,2k)*C(n-k,k)*(-1)^(n-k). - Zhi-Wei Sun, Nov 12 2016

Example

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.

Mathematica

s[n_]:=Sum[Binomial[n, k]^2*Binomial[2k, k](-1)^k, {k, 0, n}]
Table[s[n], {n, 0, 20}]

Prog

(PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n, k)^2*binomial(2*k, k)); \\ Michel Marcus, Nov 13 2016

Crossrefs

Cf. A000172, A002893, A275027, A277640.

Sequence in context: A010005 A172076 A063522 * A145850 A125992 A363714

Adjacent sequences: A244970 A244971 A244972 * A244974 A244975 A244976

Keyword

sign

Author

Zhi-Wei Sun, Jul 08 2014

Status

approved