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A157012 Riordan's general Eulerian recursion: T(n,k) = (k+2)*T(n-1, k) + (n-k) * T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0.

%I A157012 #5 Feb 22 2019 17:34:15
%S A157012 1,1,0,1,1,0,1,5,1,0,1,18,14,1,0,1,58,110,33,1,0,1,179,672,495,72,1,0,
%T A157012 1,543,3583,5163,1917,151,1,0,1,1636,17590,43730,32154,6808,310,1,0,1,
%U A157012 4916,81812,324190,411574,176272,22904,629,1,0
%N A157012 Riordan's general Eulerian recursion: T(n,k) = (k+2)*T(n-1, k) + (n-k) * T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0.
%C A157012 Row sums are:
%C A157012 {1, 1, 2, 7, 34, 203, 1420, 11359, 102230, 1022299,...}.
%C A157012 This recursion set doesn't seem to produce the Eulerian 2nd A008517.
%C A157012 The Mathematica code gives ten sequences of which the first few are in the OEIS (see Crossrefs section). - _G. C. Greubel_, Feb 22 2019
%D A157012 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215
%H A157012 G. C. Greubel, <a href="/A157012/b157012.txt">Rows n=0..100 of triangle, flattened</a>
%F A157012 e(n,k,m) = (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) for m=2.
%F A157012 T(n,k) = (k+2)*T(n-1, k) + (n-k)*T(n-1, k-1), with T(n,0) = 1, T(n,n) = 0. - _G. C. Greubel_, Feb 22 2019
%e A157012 Triangle begins with:
%e A157012 1.
%e A157012 1,    0.
%e A157012 1,    1,     0.
%e A157012 1,    5,     1,      0.
%e A157012 1,   18,    14,      1,      0.
%e A157012 1,   58,   110,     33,      1,      0.
%e A157012 1,  179,   672,    495,     72,      1,     0.
%e A157012 1,  543,  3583,   5163,   1917,    151,     1,   0.
%e A157012 1, 1636, 17590,  43730,  32154,   6808,   310,   1,   0.
%e A157012 1, 4916, 81812, 324190, 411574, 176272, 22904, 629,   1,   0.
%t A157012 e[n_, 0, m_]:= 1;
%t A157012 e[n_, k_, m_]:= 0 /; k >= n;
%t A157012 e[n_, k_, m_]:= (k+m)*e[n-1, k, m] +(n-k+1-m)*e[n-1, k-1, m];
%t A157012 Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
%t A157012 T[n_, 0]:= 1; T[n_, n_]:= 0; T[n_, k_]:= T[n, k] = (k+2)*T[n-1, k] +(n-k) *T[n-1, k-1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Feb 22 2019 *)
%o A157012 (PARI) {T(n, k) = if(k==0, 1, if(k==n, 0, (k+2)*T(n-1, k) + (n-k)* T(n-1, k-1)))};
%o A157012 for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ _G. C. Greubel_, Feb 22 2019
%o A157012 (Sage)
%o A157012 def T(n, k):
%o A157012     if (k==0): return 1
%o A157012     elif (k==n): return 0
%o A157012     else: return (k+2)*T(n-1, k) + (n-k)* T(n-1, k-1)
%o A157012 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Feb 22 2019
%Y A157012 Cf. A008517.
%Y A157012 Cf. A157011 (m=0), A008292 (m=1), This Sequence (m=2), A157013 (m=3).
%K A157012 nonn,tabl
%O A157012 0,8
%A A157012 _Roger L. Bagula_, Feb 21 2009