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A073474 Triangle T(n,k) read by rows, where o.g.f. for T(n,k) is n!*Sum_{k=0..n} (1+x)^(n-k)/k!.

%I A073474 #18 Oct 16 2024 10:33:53
%S A073474 1,2,1,5,6,2,16,33,24,6,65,196,228,120,24,326,1305,2120,1740,720,120,
%T A073474 1957,9786,20550,23160,14760,5040,720,13700,82201,212352,305970,
%U A073474 265440,138600,40320,5040,109601,767208,2356424,4146576,4571280,3232320,1431360,362880,40320
%N A073474 Triangle T(n,k) read by rows, where o.g.f. for T(n,k) is n!*Sum_{k=0..n} (1+x)^(n-k)/k!.
%C A073474 Row sums give A010844.
%H A073474 Alois P. Heinz, <a href="/A073474/b073474.txt">Rows n = 0..140, flattened</a>
%F A073474 E.g.f.: exp(x)/(1-x-x*y). - _Vladeta Jovovic_, Oct 17 2003
%F A073474 T(n, k) = Sum_{j=0..n} binomial(j, k)*FallingFactorial(n, j). - _Peter Luschny_, Oct 16 2024
%e A073474 Triangle begins:
%e A073474     1;
%e A073474     2,    1;
%e A073474     5,    6,    2;
%e A073474    16,   33,   24,    6;
%e A073474    65,  196,  228,  120,  24;
%e A073474   326, 1305, 2120, 1740, 720, 120;
%e A073474   ...
%p A073474 G:=simplify(series(exp(x)/(1-x-x*y),x=0,13)): P[0]:=1: for n from 1 to 11 do P[n]:=sort(n!*coeff(G,x^n)) od: seq(seq(coeff(y*P[n],y^k),k=1..n+1),n=0..9);
%p A073474 # second Maple program:
%p A073474 b:= proc(n, k) option remember; `if`(k>n, 0, `if`(k=0, 1,
%p A073474       n*(b(n-1, k-1)+b(n-1, k))))
%p A073474     end:
%p A073474 T:= (n, k)-> b(n+1, k+1)/(n+1):
%p A073474 seq(seq(T(n, k), k=0..n), n=0..9);  # _Alois P. Heinz_, Sep 12 2019
%t A073474 b[n_, k_] := b[n, k] = If[k>n, 0, If[k==0, 1, n (b[n-1, k-1]+b[n-1, k])]];
%t A073474 T[n_, k_] := b[n+1, k+1]/(n+1);
%t A073474 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 09 2019, after _Alois P. Heinz_ *)
%t A073474 T[n_, k_] := Sum[Binomial[j, k] FactorialPower[n, j], {j, 0, n}]; (* _Peter Luschny_, Oct 16 2024 *)
%o A073474 (SageMath)
%o A073474 def T(n, k): return sum(binomial(j, k) * falling_factorial(n, j) for j in range(n+1))
%o A073474 for n in range(8): print([T(n, k) for k in range(n+1)])
%o A073474 # _Peter Luschny_, Oct 16 2024
%Y A073474 Cf. A000142, A000522, A073107, A010844 (row sums).
%K A073474 easy,nonn,tabl
%O A073474 0,2
%A A073474 _Vladeta Jovovic_, Aug 26 2002
%E A073474 Edited by _Emeric Deutsch_, Jun 10 2004