A093963 - OEIS

%I #9 Dec 30 2021 01:28:50

%S 1,3,8,20,49,123,312,824,2221,6235,17904,53348,162545,511747,1645776,

%T 5448600,18404189,63794611,225353368,814801812,2999022641,11274044075,

%U 43100574472,167987074584,665229445293,2681607587627,10973746015456

%N Antidiagonal sums of array in A093966.

%H G. C. Greubel, <a href="/A093963/b093963.txt">Table of n, a(n) for n = 1..875</a>

%F 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

%t 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}]]]];

%t a[n_]:= a[n]= Sum[A[k, n-k+1], {k, n}];

%t Table[a[n], {n, 30}] (* _G. C. Greubel_, Dec 29 2021 *)

%o (Sage)

%o @CachedFunction

%o def A(n, k):

%o     if (n==1): return 1

%o     elif (k==1): return n

%o     elif (2 <= k < n+1): return factorial(k)*binomial(n, k) + sum( j*factorial(j)*binomial(n, j) for j in (1..k-1) )

%o     else: return sum( j*factorial(j)*binomial(n, j) for j in (1..n) )

%o @CachedFunction

%o def a(n): return sum( A(k, n-k+1) for k in (1..n) )

%o [a(n) for n in (1..30)] # _G. C. Greubel_, Dec 29 2021

%Y Cf. A093964, A093965.

%K nonn

%O 1,2

%A _Ralf Stephan_, Apr 20 2004