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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. 0
5, 9, 69, 407, 2997, 22005, 169389, 1325889, 10573677, 85386881, 697013325, 5739021051, 47599593941, 397234035333, 3332690347437, 28089543969855, 237711099004461, 2018856328439841, 17200553934626253, 146966002696538271 (list; graph; refs; listen; history; text; internal format)
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: A222536 A222698 A200440 * A304538 A306123 A192202

Adjacent sequences:  A179097 A179098 A179099 * A179101 A179102 A179103

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jun 29 2010

STATUS

approved

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Last modified April 18 15:54 EDT 2021. Contains 343089 sequences. (Running on oeis4.)