OFFSET
1,2
COMMENTS
Conjecture: All the terms are integers.
This is motivated by Conjecture 4.13 and Remark 4.13 in the linked 2023 paper of Z.-W. Sun.
LINKS
Zhi-Wei Sun, New congruences involving harmonic numbers, Nanjing Univ. J. Math. Biquarterly 40 (2023), 1-33.
FORMULA
a(n) ~ 3^(3*n - 1/2) / (20*Pi*n^2). - Vaclav Kotesovec, Apr 01 2024
EXAMPLE
a(2) = 3 since (145*0^2+104*0+18)*C(2*0,0)*C(3*0,0)^2/(2*0+1) + (145*1^2+104*1+18)*C(2*1,1)*C(3*1,1)^2/(2*1+1) divided by 6*2*(2*2-1)*C(3*2,2) coincides with (18+267*2*3^2/3)/(36*15) = 3.
MATHEMATICA
a[n_]:=a[n]=Sum[(145k^2+104k+18)Binomial[2k, k]Binomial[3k, k]^2/(2k+1), {k, 0, n-1}]/(6n*(2n-1)Binomial[3n, n]); Table[a[n], {n, 1, 20}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 01 2024
STATUS
approved