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A326659 T(n,k) = [0 and n>=0]; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

%I A326659 #40 Feb 09 2021 09:39:58
%S A326659 1,1,1,1,4,2,1,15,18,6,1,64,132,96,24,1,325,980,1140,600,120,1,1956,
%T A326659 7830,12720,10440,4320,720,1,13699,68502,143850,162120,103320,35280,
%U A326659 5040,1,109600,657608,1698816,2447760,2123520,1108800,322560,40320
%N A326659 T(n,k) = [0<k<=n] * n*(T(n-1,k-1)+T(n-1,k)) + [k=0 and n>=0]; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.
%C A326659 [] is an Iverson bracket.
%H A326659 Alois P. Heinz, <a href="/A326659/b326659.txt">Rows n = 0..140, flattened</a>
%H A326659 Wikipedia, <a href="https://en.wikipedia.org/wiki/Iverson_bracket">Iverson bracket</a>
%F A326659 E.g.f. of column k: exp(x)*(x/(1-x))^k.
%F A326659 T(n,k) = k! * A271705(n,k).
%F A326659 T(n,k) = n * A073474(n-1,k-1) for n,k >= 1.
%F A326659 T(n,1) = n * A000522(n-1) for n >= 1.
%F A326659 T(n,2) = n * A093964(n-1) for n >= 1.
%F A326659 Sum_{k=1..n} k * T(n,k) = A327606(n).
%e A326659 Triangle T(n,k) begins:
%e A326659   1;
%e A326659   1,     1;
%e A326659   1,     4,     2;
%e A326659   1,    15,    18,      6;
%e A326659   1,    64,   132,     96,     24;
%e A326659   1,   325,   980,   1140,    600,    120;
%e A326659   1,  1956,  7830,  12720,  10440,   4320,   720;
%e A326659   1, 13699, 68502, 143850, 162120, 103320, 35280, 5040;
%e A326659   ...
%p A326659 T:= proc(n, k) option remember;
%p A326659       `if`(0<k and k<=n, n*(T(n-1, k-1)+T(n-1, k)), 0)+
%p A326659       `if`(k=0 and n>=0, 1, 0)
%p A326659     end:
%p A326659 seq(seq(T(n, k), k=0..n), n=0..10);
%t A326659 T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = Boole[0 < k <= n]*n*(T[n-1, k-1] + T[n-1, k]) + Boole[k == 0 && n >= 0];
%t A326659 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 09 2021 *)
%Y A326659 Columns k=0-2 give: A000012, A007526, 2*A134432(n-1).
%Y A326659 Main diagonal gives A000142.
%Y A326659 Row sums give A308876.
%Y A326659 Cf. A000522, A073474, A093964, A143409, A196347, A271705, A327606.
%K A326659 nonn,tabl
%O A326659 0,5
%A A326659 _Alois P. Heinz_, Sep 12 2019