A093963 Antidiagonal sums of array in A093966.

1, 3, 8, 20, 49, 123, 312, 824, 2221, 6235, 17904, 53348, 162545, 511747, 1645776, 5448600, 18404189, 63794611, 225353368, 814801812, 2999022641, 11274044075, 43100574472, 167987074584, 665229445293, 2681607587627, 10973746015456

Offset: 1

Author

Ralf Stephan, Apr 20 2004

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..875

Crossrefs

Cf. A093964, A093965.

Programs

Mathematica

A[n_, k_]:= A[n, k]= If[n==1, 1, If[k==1, n, If[2<=kG. C. Greubel, Dec 29 2021 *)

Sage

@CachedFunction
def A(n, k):
    if (n==1): return 1
    elif (k==1): return n
    elif (2 <= k < n+1): return factorial(k)*binomial(n, k) + sum( j*factorial(j)*binomial(n, j) for j in (1..k-1) )
    else: return sum( j*factorial(j)*binomial(n, j) for j in (1..n) )
@CachedFunction
def a(n): return sum( A(k, n-k+1) for k in (1..n) )
[a(n) for n in (1..30)] # G. C. Greubel, Dec 29 2021

Formula

Conjecture: 2*a(n) -5*a(n-1) -(n+2)*a(n-2) +2*(n+6)*a(n-3) +(n-13)*a(n-4) -4*(n-3)*a(n-5) +2*(n-3)*a(n-6) = 0. - R. J. Mathar, Nov 10 2013