OFFSET
0,2
COMMENTS
Equals row sums of triangle A396840.
Sum_{n>=0} a(n)/H(n) = 5.10441588436913845098284370427369710398865105633585...
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..60
FORMULA
G.f. A(x) = ( Sum_{n>=0} x^n/H(n) )^2 = Sum_{n>=0} a(n)*x^n/H(n), where H(n) = Product_{k=0..n} k^k is the hyperfactorial of n (A002109).
a(n) = Sum_{k=0..n} H(n)/(H(k)*H(n-k)) for n >= 0, where H(n) = A002109(n).
a(n) = Sum_{k=0..n} Product_{j=1..k} (n+1-j)^(n+1-j) / j^j for n >= 0.
a(n) ~ 2^(n^2/4 + n/2 + 1/6) * n^(n^2/4 - 1/12) / (A*exp(n^2/8)) if n is even and a(n) ~ 2^(n^2/4 + n/2 + 17/12) * n^(n^2/4 - 1/3) / (A*exp(n^2/8 + 1/4)) if n is odd, where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jul 06 2026
EXAMPLE
G.f. A(x) = 1 + 2*x/H(1) + 6*x^2/H(2) + 56*x^3/H(3) + 2242*x^4/H(4) + 406252*x^5/H(5) + 418593314*x^6/H(6) + 2242779358192*x^7/H(7) + 84806401977745410*x^8/H(8) + ...
where denominators are hyperfactorial numbers H(n) = A002109(n).
Let E(x) = Sum_{n>=0} x^n/H(n) = 1 + x + x^2/4 + x^3/108 + x^4/27648 + x^5/86400000 + x^6/4031078400000 + x^7/3319766398771200000 + ...
then the g.f. may be expressed by
A(x) = E(x)^2 = 1 + (1 + 1)*x^2/H(1) + (1 + 4 + 1)*x^2/H(2) + (1 + 27 + 27 + 1)*x^3/H(3) + (1 + 256 + 1728 + 256 + 1)*x^4/H(4) + (1 + 3125 + 200000 + 200000 + 3125 + 1)*x^5/H(5) + ... + (Sum_{k=0..n} H(n)/(H(k)*H(n-k)))*x^n/H(n) + ...
the terms a(n) thus equal the row sums of triangle A396840.
MATHEMATICA
Table[Sum[Hyperfactorial[n] / (Hyperfactorial[k]*Hyperfactorial[n-k]), {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 06 2026 *)
PROG
(PARI) \\ by definition
{H(n) = prod(k=1, n, k^k)} \\ hyperfactorial of n
{a(n) = sum(k=0, n, H(n)/(H(k)*H(n-k)) )}
for(n=0, 15, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 05 2026
STATUS
approved