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%I #29 Mar 15 2022 03:04:19
%S 1,1,1,1,8,1,1,39,39,1,1,166,546,166,1,1,677,5482,5482,677,1,1,2724,
%T 47175,109640,47175,2724,1,1,10915,373809,1709675,1709675,373809,
%U 10915,1,1,43682,2824048,23077694,44451550,23077694,2824048,43682,1
%N Triangle T(n,k) read by rows: T(n,k) = 1 if k=1 or k=n, otherwise T(n,k) = (3*n-3*k+1)*T(n-1,k-1) + (3*k-2)*T(n-1,k).
%C Consider the triangle T(n,k) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k). For m = ...,-2,-1,0,1,2,3,... we get ..., A225372, A144431, A007318, A008292, A060187, A142458, ... - _N. J. A. Sloane_, May 08 2013
%H G. C. Greubel, <a href="/A142458/b142458.txt">Rows n = 1..50 of the triangle, flattened</a>
%H G. Strasser, <a href="http://dx.doi.org/10.1017/S0305004110000538">Generalisation of the Euler adic</a>, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_3(n,k).
%F T(n, k) = (m*n-m*k+1)*T(n-1,k-1) + (m*k-m+1)*T(n-1,k), with T(n, 1) = T(n, n) = 1, and m = 3.
%F Sum_{k=1..n} T(n, k) = A008544(n-1).
%F From _G. C. Greubel_, Mar 14 2022: (Start)
%F T(n, n-k) = T(n, k).
%F T(n, 2) = A144414(n-1).
%F T(n, 3) = A142976(n-2).
%F T(n, 4) = A144380(n-3).
%F T(n, 5) = A144381(n-4). (End)
%e The rows n >= 1 and columns 1 <= k <= n look as follows:
%e 1;
%e 1, 1;
%e 1, 8, 1;
%e 1, 39, 39, 1;
%e 1, 166, 546, 166, 1;
%e 1, 677, 5482, 5482, 677, 1;
%e 1, 2724, 47175, 109640, 47175, 2724, 1;
%e 1, 10915, 373809, 1709675, 1709675, 373809, 10915, 1;
%e 1, 43682, 2824048, 23077694, 44451550, 23077694, 2824048, 43682, 1;
%p A142458 := proc(n,k) if n = k then 1; elif k > n or k < 1 then 0 ;else (3*n-3*k+1)*procname(n-1,k-1)+(3*k-2)*procname(n-1,k) ; end if; end proc:
%p seq(seq(A142458(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Jun 04 2011
%t T[n_, k_, m_]:= T[n, k, m]= If[k==1 || k==n, 1, (m*n-m*k+1)*T[n-1, k-1, m] + (m*k -m+1)*T[n-1, k, m] ];
%t Table[T[n, k, 3], {n, 1, 10}, {k, 1, n}]//Flatten (* modified by _G. C. Greubel_, Mar 14 2022 *)
%o (Sage)
%o def T(n,k,m): # A142458
%o if (k==1 or k==n): return 1
%o else: return (m*(n-k)+1)*T(n-1,k-1,m) + (m*k-m+1)*T(n-1,k,m)
%o flatten([[T(n,k,3) for k in (1..n)] for n in (1..10)]) # _G. C. Greubel_, Mar 14 2022
%Y Cf. A225372 (m=-2), A144431 (m=-1), A007318 (m=0), A008292 (m=1), A060187 (m=2), this sequence (m=3), A142459 (m=4), A142560 (m=5), A142561 (m=6), A142562 (m=7), A167884 (m=8), A257608 (m=9).
%Y Cf. A008544, A142976, A144380, A144381, A144414.
%K nonn,easy,tabl
%O 1,5
%A _Roger L. Bagula_, Sep 19 2008
%E Edited by the Associate Editors of the OEIS, Aug 28 2009
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