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%I #8 Jun 10 2021 02:56:15
%S 1,0,1,0,0,4,0,3,0,27,0,8,96,0,256,0,45,640,2430,0,3125,0,264,8640,
%T 29160,61440,0,46656,0,1855,118272,688905,1146880,1640625,0,823543,0,
%U 14832,1899520,16166304,41287680,43750000,47029248,0,16777216,0,133497,34172928,438143580,1453326336,2214843750,1693052928,1452729852,0,387420489
%N Triangle T(n, k) = binomial(n, k)*A000166(n-k)*k^n with T(0, 0) = 1, read by rows.
%D J. Riordan, Combinatorial Identities, Wiley, 1968, p.194.
%H G. C. Greubel, <a href="/A156788/b156788.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = binomial(n, k)*A000166(n-k)*k^n with T(0, 0) = 1.
%F T(n, k) = binomial(n, k)*b(n-k)*k^n, where b(n) = n*b(n-1) + (-1)^n and b(0) = 1.
%F Sum_{k=0..n} T(n, k) = A137341(n).
%F From _G. C. Greubel_, Jun 10 2021: (Start)
%F T(n, 1) = A000240(n).
%F T(n, n) = A000312(n). (End)
%e Triangle begins as:
%e 1;
%e 0, 1;
%e 0, 0, 4;
%e 0, 3, 0, 27;
%e 0, 8, 96, 0, 256;
%e 0, 45, 640, 2430, 0, 3125;
%e 0, 264, 8640, 29160, 61440, 0, 46656;
%e 0, 1855, 118272, 688905, 1146880, 1640625, 0, 823543;
%e 0, 14832, 1899520, 16166304, 41287680, 43750000, 47029248, 0, 16777216;
%t A000166[n_]:= A000166[n]= If[n==0, 1, n*A000166[n-1] + (-1)^n];
%t T[n_, k_]:= If[n==0, 1, Binomial[n, k]*A000166[n-k]*k^n];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jun 10 2021 *)
%o (Sage)
%o def A000166(n): return 1 if (n==0) else n*A000166(n-1) + (-1)^n
%o def A156788(n,k): return 1 if (n==0) else binomial(n,k)*k^n*A000166(n-k)
%o flatten([[A156788(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 10 2021
%Y Cf. A000166, A000240, A000312, A137341.
%K nonn,tabl
%O 0,6
%A _Roger L. Bagula_, Feb 15 2009
%E Edited by _G. C. Greubel_, Jun 10 2021
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