OFFSET
0,2
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.
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;
MATHEMATICA
Table[Binomial[n*(n+3)/2, n], {n, 0, 40}] (* G. C. Greubel, Feb 19 2022 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 04 2005
STATUS
approved