

A278415


a(n) = Sum_{k=0..n} binomial(n, 2k)*binomial(nk, k)*(1)^k.


2



1, 1, 0, 5, 16, 24, 15, 197, 576, 724, 1200, 8832, 22801, 21293, 76440, 408795, 922368, 499104, 4446588, 19025060, 37012416, 1673992, 245604832, 880263936, 1441226991, 908700649, 13088509200, 40222012703, 52991533744, 88167061704, 678172355415, 1805175708261, 1747974632448, 6237554623536, 34300087628480
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OFFSET

0,4


COMMENTS

Conjecture: For any prime p > 3 and positive integer n, the number (a(p*n)a(n))/(p*n)^2 is always a padic integer.
We are able to show that for any prime p > 3 and positive integer n the number (a(p*n)a(n))/(p^2*n) is always a padic integer.
See also A275027 and A278405 for similar conjectures.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..400
ZhiWei Sun, Supercongruences involving Lucas sequences, arXiv:1610.03384 [math.NT], 2016.


EXAMPLE

a(3) = 5 since a(3) = C(3, 2*0)*C(30, 0)(1)^0 + C(3,2*1)*C(31,1)(1)^1 = 1  6 = 5.


MATHEMATICA

a[n_]:=Sum[Binomial[n, 2k]Binomial[nk, k](1)^k, {k, 0, n}]
Table[a[n], {n, 0, 34}]


CROSSREFS

Cf. A208425, A244973, A277640, A275027, A278405.
Sequence in context: A029450 A225539 A299124 * A063243 A063232 A087747
Adjacent sequences: A278412 A278413 A278414 * A278416 A278417 A278418


KEYWORD

sign


AUTHOR

ZhiWei Sun, Nov 21 2016


STATUS

approved



