A179100 a(n) = (1/n) * Sum_{k=0..n-1} (8k+5) T_k^2, where T_0, T_1, ... are central trinomial coefficients given by A002426.

5, 9, 69, 407, 2997, 22005, 169389, 1325889, 10573677, 85386881, 697013325, 5739021051, 47599593941, 397234035333, 3332690347437, 28089543969855, 237711099004461, 2018856328439841, 17200553934626253, 146966002696538271

Offset

1,1

Comments

On Jun 17 2010, Zhi-Wei Sun conjectured that a(n) is an integer for every n=1,2,3,... and that a(p) == 3(p/3) (mod p) for any prime p, where (p/3) is the Legendre symbol. He also observed that Sum_{k=0..n-1} (2k+1) T_k*3^{n-1-k} = n * Sum_{k=0..n-1} C(n-1,k)*(-1)^(n-1-k)*(k+1)*C(2k,k).

Links

Table of n, a(n) for n=1..20.

Zhi-Wei Sun, Arithmetic properties of Apery numbers and central Delannoy numbers, preprint, arXiv:1006.2776 [math.NT], 2010-2011.

Example

For n=3 we have a(3) = (5*T_0^2 + 13*T_1^2 + 21*T_2^2)/3 = (5 + 13 + 21*9)/3 = 69.

Mathematica

TT[n_]:=Sum[Binomial[n, 2k]Binomial[2k, k], {k, 0, Floor[n/2]}] SS[n_]:=Sum[(8k+5)*TT[k]^2, {k, 0, n-1}]/n Table[SS[n], {n, 1, 50}]

Crossrefs

Cf. A002426, A179089, A178808, A178790, A178791, A173774.

Sequence in context: A200440 A370613 A347991 * A358966 A304538 A306123

Adjacent sequences: A179097 A179098 A179099 * A179101 A179102 A179103

Keyword

nonn

Author

Zhi-Wei Sun, Jun 29 2010

Status

approved