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A143122 Triangle read by rows, T(n,k) = Sum_{j=k..n} j!, 0 <= k <= n.

%I A143122 #20 Jul 11 2024 08:37:51
%S A143122 1,2,1,4,3,2,10,9,8,6,34,33,32,30,24,154,153,152,150,144,120,874,873,
%T A143122 872,870,864,840,720,5914,5913,5912,5910,5904,5880,5760,5040,46234,
%U A143122 46233,46232,46230,46224,46200,46080,45360,40320
%N A143122 Triangle read by rows, T(n,k) = Sum_{j=k..n} j!, 0 <= k <= n.
%C A143122 Left column = A003422 starting (1, 2, 4, 10, 34, ...).
%C A143122 Row sums = A007489 starting (1, 3, 9, 33, 153, ...).
%H A143122 G. C. Greubel, <a href="/A143122/b143122.txt">Rows n = 0..100 of triangle, flattened</a>
%F A143122 Triangle read by rows, T(n,k) = Sum_{j=k..n} j!, 0 <= k <= n.
%F A143122 A000012 * (A000142 * 0^(n-k)) * A000012, where A000142 = (1, 1, 2, 6, ...).
%F A143122 (The above is to be read as matrix product A*D*A where A is lower triangular filled with 1's and D = diag(n!, n >= 0). - _M. F. Hasler_, Aug 26 2020)
%F A143122 T(n, k) = t(k) - t(n + 1), where t(n) = (-1)^(n + 1)*Gamma(n + 1)*Subfactorial(-(n + 1)). - _Peter Luschny_, Jul 11 2024
%e A143122 First few rows of the triangle are:
%e A143122    1;
%e A143122    2,  1;
%e A143122    4,  3,  2;
%e A143122   10,  9,  8,  6;
%e A143122   34, 33, 32, 30, 24;
%e A143122   ...
%e A143122 T(4,2) = 32 = 4! + 3! + 2! = (24 + 6 + 2).
%p A143122 a:=proc(n,k) local j; add(factorial(j),j=k..n) end: seq(seq(a(n,k),k=0..n),n=0..8); # _Muniru A Asiru_, Oct 16 2018
%t A143122 Table[Sum[j!, {j, k, n}], {n, 0, 15}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Oct 15 2018 *)
%o A143122 (PARI) for(n=0,15, for(k=0,n, print1(sum(j=k,n, j!), ", "))) \\ _G. C. Greubel_, Oct 15 2018
%o A143122 (PARI) (A143122(n,k)=sum(j=k,n, j!)); T(n)=matrix(n++,n,i,j,i>=j)*matrix(n,n,i,j,(i>=j)*i--!) \\ _M. F. Hasler_, Aug 26 2020
%o A143122 (Magma) [[(&+[Factorial(j): j in [k..n]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Oct 15 2018
%o A143122 (GAP) Flat(List([0..8],n->List([0..n],k->Sum([k..n],j->Factorial(j))))); # _Muniru A Asiru_, Oct 16 2018
%Y A143122 Cf. A000142, A003422, A007489.
%K A143122 nonn,tabl
%O A143122 0,2
%A A143122 _Gary W. Adamson_ & _Roger L. Bagula_, Jul 26 2008