A135860 a(n) = binomial(n*(n+1), n).

1, 2, 15, 220, 4845, 142506, 5245786, 231917400, 11969016345, 706252528630, 46897636623981, 3461014728350400, 281014969393251275, 24894763097057357700, 2389461906843449885700, 247012484980695576597296, 27361230617617949782033713, 3233032526324680287912449550

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..337

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k). - Paul D. Hanna, Nov 18 2015

a(n) is divisible by (n+1): a(n)/(n+1) = A135861(n).

a(n) is divisible by (n^2+1): a(n)/(n^2+1) = A135862(n).

a(n) = binomial(2*A000217(n),n). - Arkadiusz Wesolowski, Jul 18 2012

a(n) = [x^n] 1/(1 - x)^(n^2+1). - Ilya Gutkovskiy, Oct 03 2017

a(n) ~ exp(n + 1/2) * n^(n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Feb 08 2019

a(p) == p + 1 ( mod p^4 ) for prime p >= 5 and a(2*p) == (4*p + 1)*(2*p + 1) ( mod p^4 ) for all prime p. Apply Mestrovic, equation 37. - Peter Bala, Feb 27 2020

a(n) = ((n^2 + n)!)/((n^2)! * n!). - Peter Luschny, Feb 27 2020

Mathematica

Table[Binomial[n^2 + n, n], {n, 0, 16}] (* Arkadiusz Wesolowski, Jul 18 2012 *)

Prog

(PARI) a(n)=binomial(n*(n+1), n)
for(n=0, 15, print1(a(n), ", "))
(PARI) a(n)=sum(k=0, n, binomial(n, k)*binomial(n^2, k))
for(n=0, 15, print1(a(n), ", "))
(Magma) [Binomial(n*(n+1), n): n in [0..30]]; // G. C. Greubel, Feb 20 2022
(Sage) [binomial(n*(n+1), n) for n in (0..30)] # G. C. Greubel, Feb 20 2022

Crossrefs

Cf. A014062, A014068, A054688, A107863, A135861, A135862.

Sequence in context: A298692 A361617 A132493 * A178533 A087962 A140054

Adjacent sequences: A135857 A135858 A135859 * A135861 A135862 A135863

Keyword

easy,nonn

Author

Paul D. Hanna, Dec 02 2007

Status

approved