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%I #13 Aug 29 2023 14:11:21
%S 1,2,4,12,72,1010,36302,3501500,984382830,820391106394,
%T 2231837962830894,19443994569352596154,611248544067759392038426,
%U 65374059149370152526265388842,27613396368509694864033710442373202
%N Sum of entries in n-th antidiagonal in A060854.
%H Alois P. Heinz, <a href="/A060935/b060935.txt">Table of n, a(n) for n = 1..40</a>
%F a(n) ~ c(n) * sqrt(Pi) * exp(7/12 + n/2 + n^2/8) * n^(11/12 + n/2 + n^2/4) / (A * 2^(5/6 + 3*n/2 + 3*n^2/4)), where c(n) = 2 if n is even and c(n) = (n/2)^(1/4) if n is odd, A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Aug 29 2023
%p T:= (m, n)-> (m*n)! * mul(i!/(m+i)!, i=0..n-1):
%p a:= n-> add (T(k, 1+n-k), k=1..n):
%p seq (a(n), n=1..20); # _Alois P. Heinz_, Aug 06 2012
%t A060854[n_, k_]:= (k*(n-k+1))!*BarnesG[k+1]*BarnesG[n-k+2]/BarnesG[n+2];
%t Table[Sum[A060854[n, k], {k,n}], {n,20}] (* _G. C. Greubel_, Apr 07 2021 *)
%o (Magma)
%o A060854:= func< n,k | Factorial((n-k+1)*k)*(&*[ Factorial(j)/Factorial(n-k+j+1): j in [0..k-1] ]) >;
%o [(&+[ A060854(n, k): k in [1..n] ]): n in [1..20]]; // _G. C. Greubel_, Apr 07 2021
%o (Sage)
%o def A060854(n, k): return factorial((n-k+1)*k)*product( factorial(j)/factorial(n-k+j+1) for j in (0..k-1) )
%o def A060935(n): return sum( A060854(n, k) for k in (1..n) )
%o [A060935(n) for n in (1..20)] # _G. C. Greubel_, Apr 07 2021
%Y Cf. A060854.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, May 06 2001
%E More terms from _Frank Ellermann_, Jun 15 2001
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