A107863 Column 1 of triangle A107862; a(n) = binomial(n*(n+1)/2 + n, n).

1, 2, 10, 84, 1001, 15504, 296010, 6724520, 177232627, 5317936260, 179013799328, 6681687099710, 273897571557780, 12233149001721760, 591315394579074378, 30756373941461374800, 1712879663609111933495, 101696990867999141755140

Offset: 0

Author

Paul D. Hanna, Jun 04 2005

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..350
• R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.

Crossrefs

Cf. A014068, A107862, A135860, A135861.

Programs

Mathematica

Table[Binomial[n*(n+3)/2, n], {n,0,40}] (* G. C. Greubel, Feb 19 2022 *)

PARI

a(n)=binomial(n*(n+1)/2+n,n)

Sage

[binomial(n*(n+3)/2, n) for n in (0..40)] # G. C. Greubel, Feb 19 2022

Formula

a(n) = [x^(n*(n+1)/2)] 1/(1 - x)^(n+1). - Ilya Gutkovskiy, Oct 10 2017

From Peter Bala, Feb 23 2020: (Start)

Put b(n) = a(n-1). We have the congruences:

b(p) == 1 (mod p^3) for prime p >= 5 (uses Mestrovic, equation 35);

b(2*p) == 2*p (mod p^4) for prime p >= 5 (uses Mestrovic, equation 44 and the von Staudt-Clausen theorem).

Conjectural congruences:

b(3*p) == (81*p*2 - 1)/8 (mod p^3) for prime p >= 3;

3*b(4*p) == -4*p (mod p^3) for all prime p. Cf. A135860 and A135861. (End)