A060935 Sum of entries in n-th antidiagonal in A060854.

1, 2, 4, 12, 72, 1010, 36302, 3501500, 984382830, 820391106394, 2231837962830894, 19443994569352596154, 611248544067759392038426, 65374059149370152526265388842, 27613396368509694864033710442373202

Offset: 1

Author

N. J. A. Sloane, May 06 2001

Keywords

• nonn
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..40

Crossrefs

Cf. A060854.

Programs

Magma

A060854:= func< n,k | Factorial((n-k+1)*k)*(&*[ Factorial(j)/Factorial(n-k+j+1): j in [0..k-1] ]) >;
[(&+[ A060854(n, k): k in [1..n] ]): n in [1..20]]; // G. C. Greubel, Apr 07 2021

Maple

T:= (m, n)-> (m*n)! * mul(i!/(m+i)!, i=0..n-1):
a:= n-> add (T(k, 1+n-k), k=1..n):
seq (a(n), n=1..20);  # Alois P. Heinz, Aug 06 2012

Mathematica

A060854[n_, k_]:= (k*(n-k+1))!*BarnesG[k+1]*BarnesG[n-k+2]/BarnesG[n+2];
Table[Sum[A060854[n, k], {k,n}], {n,20}] (* G. C. Greubel, Apr 07 2021 *)

Sage

def A060854(n, k): return factorial((n-k+1)*k)*product( factorial(j)/factorial(n-k+j+1) for j in (0..k-1) )
def A060935(n): return sum( A060854(n, k) for k in (1..n) )
[A060935(n) for n in (1..20)] # G. C. Greubel, Apr 07 2021

Formula

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

Extensions

More terms from Frank Ellermann, Jun 15 2001