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A093963 Antidiagonal sums of array in A093966.

%I A093963 #9 Dec 30 2021 01:28:50
%S A093963 1,3,8,20,49,123,312,824,2221,6235,17904,53348,162545,511747,1645776,
%T A093963 5448600,18404189,63794611,225353368,814801812,2999022641,11274044075,
%U A093963 43100574472,167987074584,665229445293,2681607587627,10973746015456
%N A093963 Antidiagonal sums of array in A093966.
%H A093963 G. C. Greubel, <a href="/A093963/b093963.txt">Table of n, a(n) for n = 1..875</a>
%F A093963 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 A093963 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 A093963 a[n_]:= a[n]= Sum[A[k, n-k+1], {k, n}];
%t A093963 Table[a[n], {n, 30}] (* _G. C. Greubel_, Dec 29 2021 *)
%o A093963 (Sage)
%o A093963 @CachedFunction
%o A093963 def A(n, k):
%o A093963     if (n==1): return 1
%o A093963     elif (k==1): return n
%o A093963     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 A093963     else: return sum( j*factorial(j)*binomial(n, j) for j in (1..n) )
%o A093963 @CachedFunction
%o A093963 def a(n): return sum( A(k, n-k+1) for k in (1..n) )
%o A093963 [a(n) for n in (1..30)] # _G. C. Greubel_, Dec 29 2021
%Y A093963 Cf. A093964, A093965.
%K A093963 nonn
%O A093963 1,2
%A A093963 _Ralf Stephan_, Apr 20 2004