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A076571 Binomial triangle based on factorials.

%I A076571 #16 Oct 06 2023 11:03:46
%S A076571 1,1,2,2,3,5,6,8,11,16,24,30,38,49,65,120,144,174,212,261,326,720,840,
%T A076571 984,1158,1370,1631,1957,5040,5760,6600,7584,8742,10112,11743,13700,
%U A076571 40320,45360,51120,57720,65304,74046,84158,95901,109601
%N A076571 Binomial triangle based on factorials.
%H A076571 G. C. Greubel, <a href="/A076571/b076571.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A076571 E. Biondi, L. Divieti, and G. Guardabassi, <a href="http://dx.doi.org/10.4153/CJM-1970-003-9">Counting paths, circuits, chains and cycles in graphs: A unified approach</a>, Canad. J. Math. 22 1970 22-35. See Table I.
%H A076571 D. Dumont, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05dumont.html">Matrices d'Euler-Seidel</a>, Sem. Loth. Comb. B05c (1981) 59-78.
%F A076571 T(n, k) = Sum_{j=0..k} binomial(k, j)*(n-j)!.
%F A076571 T(n, k) = T(n, k-1) + T(n-1, k-1) with T(n, 0) = n!.
%F A076571 T(n, n) = A000522(n).
%F A076571 Sum_{k=0..n} T(n, k) = A002627(n+1).
%F A076571 From _G. C. Greubel_, Oct 05 2023: (Start)
%F A076571 T(n, k) = n! * Hypergeometric1F1([-k], [-n], 1).
%F A076571 T(2*n, n) = A099022(n). (End)
%e A076571 Rows start:
%e A076571     1;
%e A076571     1,   2;
%e A076571     2,   3,   5;
%e A076571     6,   8,  11,  16;
%e A076571    24,  30,  38,  49,  65;
%e A076571   120, 144, 174, 212, 261, 326;
%t A076571 A076571[n_, k_]:= n!*Hypergeometric1F1[-k,-n,1];
%t A076571 Table[A076571[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Oct 05 2023 *)
%o A076571 (Magma)
%o A076571 A076571:= func< n,k| (&+[Binomial(k,j)*Factorial(n-j): j in [0..k]]) >;
%o A076571 [A076571(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 05 2023
%o A076571 (SageMath)
%o A076571 def A076571(n,k): return sum(binomial(k,j)*factorial(n-j) for j in range(k+1))
%o A076571 flatten([[A076571(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 05 2023
%Y A076571 Columns include A000142, A001048, A001344, A001345, A001346, A001347.
%Y A076571 Right hand columns include A000522, A001339, A001340, A001341, A001342.
%Y A076571 Cf. A002627 (row sums), A099022.
%K A076571 nonn,tabl
%O A076571 0,3
%A A076571 _Henry Bottomley_, Oct 19 2002