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%I #12 May 25 2023 03:57:32
%S 1,1,-1,1,-4,1,1,-15,5,-1,1,-58,10,-6,1,1,-229,-66,-26,7,-1,1,-912,
%T -1017,-288,23,-8,1,1,-3643,-8733,-4779,-415,-41,9,-1,1,-14566,-61880,
%U -63606,-17242,-1158,40,-10,1,1,-58257,-396796,-691036,-375118,-60990,-1956,-60,11,-1
%N Riordan's general Eulerian recursion: T(n, k) = (k+2)*T(n-1, k) + (n-k-1) * T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1).
%C Row sums are {1, 0, -2, -10, -52, -314, -2200, -17602, -158420, -1584202, ...}.
%C This recursion set doesn't seem to produce the Eulerian 2nd A008517.
%C 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 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 214-215
%H G. C. Greubel, <a href="/A157013/b157013.txt">Rows n = 1..100 of triangle, flattened</a>
%F e(n,k,m)= (k+m)*e(n-1, k, m) + (n-k+1-m)*e(n-1, k-1, m) with m=3.
%F T(n, k) = (k+2)*T(n-1, k) + (n-k-1)*T(n-1, k-1) with T(n,1) = 1, T(n,n) = (-1)^(n-1). - _G. C. Greubel_, Feb 22 2019
%e Triangle begins with:
%e 1.
%e 1, -1.
%e 1, -4, 1.
%e 1, -15, 5, -1.
%e 1, -58, 10, -6, 1.
%e 1, -229, -66, -26, 7, -1.
%e 1, -912, -1017, -288, 23, -8, 1.
%e 1, -3643, -8733, -4779, -415, -41, 9, -1.
%e 1, -14566, -61880, -63606, -17242, -1158, 40, -10, 1.
%e 1, -58257, -396796, -691036, -375118, -60990, -1956, -60, 11, -1.
%t e[n_, 0, m_]:= 1;
%t e[n_, k_, m_]:= 0 /; k >= n;
%t e[n_, k_, m_]:= (k+m)*e[n-1, k, m] + (n-k+1-m)*e[n-1, k-1, m];
%t Table[Flatten[Table[Table[e[n, k, m], {k,0,n-1}], {n,1,10}]], {m,0,10}]
%t T[n_,1]:=1; T[n_,n_]:=(-1)^(n-1); T[n_,k_]:= T[n,k] = (k+2)*T[n-1,k] + (n-k-1)*T[n-1,k-1]; Table[T[n,k], {n,1,10}, {k,1,n}]//Flatten (* _G. C. Greubel_, Feb 22 2019 *)
%o (PARI) {T(n, k) = if(k==1, 1, if(k==n, (-1)^(n-1), (k+2)*T(n-1, k) + (n-k-1)* T(n-1, k-1)))};
%o for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ _G. C. Greubel_, Feb 22 2019
%o (Sage)
%o def T(n, k):
%o if (k==1): return 1
%o elif (k==n): return (-1)^(n-1)
%o else: return (k+2)*T(n-1, k) + (n-k-1)* T(n-1, k-1)
%o [[T(n, k) for k in (1..n)] for n in (1..10)] # _G. C. Greubel_, Feb 22 2019
%Y Cf. A008517.
%Y Cf. A157011 (m=0), A008292 (m=1), A157012 (m=2), this sequence (m=3).
%K sign,tabl
%O 1,5
%A _Roger L. Bagula_, Feb 21 2009
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