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%I #18 Aug 29 2023 15:04:53
%S 1,3,9,51,795,43923,10372323,11996843043,75315947454723,
%T 2788806652875290883,654625444656522114316803,
%U 1045012738906587147509753740803,12046169853230117709495421609499289603,1053916215003128938522329980606467994425804803
%N 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).
%H G. C. Greubel, <a href="/A193521/b193521.txt">Table of n, a(n) for n = 0..48</a>
%F From _G. C. Greubel_, Jan 05 2022: (Start)
%F a(n) = Sum_{k=0..n} Sum_{j=0..k} BarnesG(n+2)/(BarnesG(j+2)*BarnesG(k-j+2 )*BarnesG(n-k+2)).
%F a(n) = Sum_{k=0..n} A009963(n, k) * Sum_{j=0..k} A009963(k, j).
%F a(n) = Sum_{j=0..n} A009963(n, j)*A193520(j). (End)
%F 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
%e Let F(x) = 1 + x + x^2/(1!*2!) + x^3/(1!*2!*3!) + x^4/(1!*2!*3!*4!) + ... + x^n/sf(n) + ...
%e 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) + ...
%t 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}];
%t Table[a[n], {n, 0, 20}] (* _G. C. Greubel_, Jan 05 2022 *)
%o (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)}
%o (Magma)
%o A193521:= func< n | (&+[ A009963(n,k)*A193520(k): k in [0..n]]) >;
%o [A193521(n): n in [0..20]]; // _G. C. Greubel_, Jan 05 2022
%o (Sage)
%o @CachedFunction
%o def A009963(n,k): return product(factorial(n-j+1)/factorial(j) for j in (1..k))
%o def A193521(n): return sum(sum(A009963(n,k)*A009963(k,j) for j in (0..k)) for k in (0..n))
%o [A193521(n) for n in (0..20)] # _G. C. Greubel_, Jan 05 2022
%Y Cf. A000178, A009963, A193520.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 29 2011
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