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A105060 Triangle read by rows in which the n-th row consists of the first n nonzero terms of A033312.

%I A105060 #18 Mar 14 2023 04:11:49
%S A105060 1,1,5,1,5,23,1,5,23,119,1,5,23,119,719,1,5,23,119,719,5039,1,5,23,
%T A105060 119,719,5039,40319,1,5,23,119,719,5039,40319,362879,1,5,23,119,719,
%U A105060 5039,40319,362879,3628799,1,5,23,119,719,5039,40319,362879,3628799,39916799
%N A105060 Triangle read by rows in which the n-th row consists of the first n nonzero terms of A033312.
%H A105060 G. C. Greubel, <a href="/A105060/b105060.txt">Rows n = 1..50 of the triangle, flattened</a>
%F A105060 From _G. C. Greubel_, Mar 13 2023: (Start)
%F A105060 T(n, k) = T(n,k-1) + k*k!, with T(n, 1) = 1.
%F A105060 Sum_{k=1..n} T(n, k) = -A007489(n+2) + (n+4)*A007489(n+1) - (n+2)*A007489(n) - (n+1). (End)
%e A105060 Triangle begins as:
%e A105060   1;
%e A105060   1, 5;
%e A105060   1, 5, 23;
%e A105060   1, 5, 23, 119;
%e A105060   1, 5, 23, 119, 719;
%e A105060   1, 5, 23, 119, 719, 5039;
%e A105060   1, 5, 23, 119, 719, 5039, 40319;
%t A105060 a[n_]:= a[n]= If[n==1, 1, a[n-1] + k!*n];
%t A105060 Table[a[k], {n,12}, {k,n}]//Flatten
%o A105060 (Magma)
%o A105060 function T(n,k)
%o A105060   if k eq 1 then return 1;
%o A105060   else return T(n,k-1) + k*Factorial(k);
%o A105060   end if;
%o A105060 end function;
%o A105060 [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 13 2023
%o A105060 (SageMath)
%o A105060 @CachedFunction
%o A105060 def T(n,k):
%o A105060     if (k==1): return 1
%o A105060     else: return T(n,k-1) + k*factorial(k)
%o A105060 flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,10)]) # _G. C. Greubel_, Mar 13 2023
%Y A105060 Cf. A007489, A033312.
%K A105060 nonn,easy,tabl
%O A105060 1,3
%A A105060 _Roger L. Bagula_, Apr 05 2005