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

1,2

Links

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

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

Mathematica

A[n_, k_]:= A[n, k]= If[n==1, 1, If[k==1, n, If[2<=k<n+1, (1-k)*k!*Binomial[n, k] + Sum[j*j!*Binomial[n, j], {j, k}], Sum[j*j!*Binomial[n, j], {j, n}]]]];
a[n_]:= a[n]= Sum[A[k, n-k+1], {k, n}];
Table[a[n], {n, 30}] (* G. C. Greubel, Dec 29 2021 *)

Prog

(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

Crossrefs

Cf. A093964, A093965.

Sequence in context: A054185 A171853 A330458 * A261233 A346944 A027219

Adjacent sequences: A093960 A093961 A093962 * A093964 A093965 A093966

Keyword

nonn

Author

Ralf Stephan, Apr 20 2004

Status

approved