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URL: https://oeis.org/A193521

⇱ A193521 - OEIS

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A193521
G.f.: A(x) = ( Sum_{n>=0} x^n/sf(n) )^3 where A(x) = Sum_{n>=0} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).
2
1, 3, 9, 51, 795, 43923, 10372323, 11996843043, 75315947454723, 2788806652875290883, 654625444656522114316803, 1045012738906587147509753740803, 12046169853230117709495421609499289603, 1053916215003128938522329980606467994425804803
OFFSET
0,2
LINKS
FORMULA
From G. C. Greubel, Jan 05 2022: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..k} BarnesG(n+2)/(BarnesG(j+2)*BarnesG(k-j+2 )*BarnesG(n-k+2)).
a(n) = Sum_{k=0..n} A009963(n, k) * Sum_{j=0..k} A009963(k, j).
a(n) = Sum_{j=0..n} A009963(n, j)*A193520(j). (End)
a(n) ~ c(n) * A^2 * 3^(5/4 + n + n^2/6) * n^(-5/6 + n^2/3) / (2*Pi * exp(1/6 + n^2/2)), where c(n) = 1 if mod(n,3) = 0 and c(n) = 3^(4/3) / n^(1/3) if mod(n,3) = 1 or if mod(n,3) = 2, A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023
EXAMPLE
Let F(x) = 1 + x + x^2/(1!*2!) + x^3/(1!*2!*3!) + x^4/(1!*2!*3!*4!) + ... + x^n/sf(n) + ...
then F(x)^3 = 1 + 3*x + 9*x^2/(1!*2!) + 51*x^3/(1!*2!*3!) + 795*x^4/(1!*2!*3!*4!) + 43923*x^5/(1!*2!*3!*4!*5!) + ... + a(n)*x^n/sf(n) + ...
MATHEMATICA
a[n_]:= a[n]= Sum[BarnesG[n+2]/(BarnesG[j+2]*BarnesG[k-j+2]*BarnesG[n-k+2]), {k, 0, n}, {j, 0, k}];
Table[a[n], {n, 0, 20}] (* G. C. Greubel, Jan 05 2022 *)
PROG
(PARI) {a(n) = prod(k=1, n, k!)*polcoeff((sum(m=0, n+1, x^m/prod(k=0, m, k!) + x*O(x^n))^3), n)}
(Magma)
A193521:= func< n | (&+[ A009963(n, k)*A193520(k): k in [0..n]]) >;
[A193521(n): n in [0..20]]; // G. C. Greubel, Jan 05 2022
(SageMath)
@CachedFunction
def A009963(n, k): return product(factorial(n-j+1)/factorial(j) for j in (1..k))
def A193521(n): return sum(sum(A009963(n, k)*A009963(k, j) for j in (0..k)) for k in (0..n))
[A193521(n) for n in (0..20)] # G. C. Greubel, Jan 05 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 29 2011
STATUS
approved