A105060 - OEIS

%I #18 Mar 14 2023 04:11:49

%S 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 119,719,5039,40319,1,5,23,119,719,5039,40319,362879,1,5,23,119,719,

%U 5039,40319,362879,3628799,1,5,23,119,719,5039,40319,362879,3628799,39916799

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

%H G. C. Greubel, <a href="/A105060/b105060.txt">Rows n = 1..50 of the triangle, flattened</a>

%F From _G. C. Greubel_, Mar 13 2023: (Start)

%F T(n, k) = T(n,k-1) + k*k!, with T(n, 1) = 1.

%F Sum_{k=1..n} T(n, k) = -A007489(n+2) + (n+4)*A007489(n+1) - (n+2)*A007489(n) - (n+1). (End)

%e Triangle begins as:

%e   1;

%e   1, 5;

%e   1, 5, 23;

%e   1, 5, 23, 119;

%e   1, 5, 23, 119, 719;

%e   1, 5, 23, 119, 719, 5039;

%e   1, 5, 23, 119, 719, 5039, 40319;

%t a[n_]:= a[n]= If[n==1, 1, a[n-1] + k!*n];

%t Table[a[k], {n,12}, {k,n}]//Flatten

%o (Magma)

%o function T(n,k)

%o   if k eq 1 then return 1;

%o   else return T(n,k-1) + k*Factorial(k);

%o   end if;

%o end function;

%o [T(n,k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Mar 13 2023

%o (SageMath)

%o @CachedFunction

%o def T(n,k):

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

%o     else: return T(n,k-1) + k*factorial(k)

%o flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,10)]) # _G. C. Greubel_, Mar 13 2023

%Y Cf. A007489, A033312.

%K nonn,easy,tabl

%O 1,3

%A _Roger L. Bagula_, Apr 05 2005