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%I #18 Feb 22 2019 21:17:57
%S 1,1,2,1,5,4,1,9,23,8,1,14,82,93,16,1,20,234,607,343,32,1,27,588,2991,
%T 3800,1189,64,1,35,1365,12501,30155,21145,3951,128,1,44,3010,47058,
%U 195626,256500,108286,12749,256,1,54,6416,165254,1111910,2456256,1932216,522387,40295,512
%N Triangle T(n,k) read by rows: T(n,k)= (k-1)*T(n-1,k) + (n-k+2)*T(n-1, k-1), with T(n,1)=1, for 1 <= k <= n, n >= 1.
%C Row sums are apparently in A002627.
%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
%H G. C. Greubel, <a href="/A157011/b157011.txt">Rows n = 1..100 of triangle, flattened</a>
%e The triangle starts in row n=1 as:
%e 1;
%e 1, 2;
%e 1, 5, 4;
%e 1, 9, 23, 8;
%e 1, 14, 82, 93, 16;
%e 1, 20, 234, 607, 343, 32;
%e 1, 27, 588, 2991, 3800, 1189, 64;
%e 1, 35, 1365, 12501, 30155, 21145, 3951, 128;
%e 1, 44, 3010, 47058, 195626, 256500, 108286, 12749, 256;
%e 1, 54, 6416, 165254, 1111910, 2456256, 1932216, 522387, 40295, 512;
%p A157011 := proc(n,k) if k <0 or k >= n then 0; elif k =0 then 1; else k*procname(n-1,k)+(n-k+1)*procname(n-1,k-1) ; end if; end proc: # _R. J. Mathar_, Jun 18 2011
%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_]:= 2^(n-1); T[n_, k_]:= T[n, k] = (k-1)*T[n-1, k] + (n-k+2)*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, 2^(n-1), (k-1)*T(n-1, k) + (n-k+2)* 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):
%o return 1
%o elif (k==n):
%o return 2^(n-1)
%o else: return (k-1)*T(n-1, k) + (n-k+2)* 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. A000096 (column k=1), A002627, A008517.
%Y Cf. This sequence (m=0), A008292 (m=1), A157012 (m=2), A157013 (m=3).
%K nonn,tabl,easy
%O 1,3
%A _Roger L. Bagula_, Feb 21 2009
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