%I #18 Feb 25 2019 18:07:51
%S 1,1,1,1,6,1,1,23,28,1,1,76,250,145,1,1,237,1608,2475,876,1,1,722,
%T 8802,26847,25056,6139,1,1,2179,43872,231057,418806,268477,49120,1,1,
%U 6552,205994,1725621,5285520,6486205,3077730,442089,1,1,19673,928808,11718015,55871814,114115195,102456300,37833831,4420900,1
%N Triangle T(n,k) = (2*n-k-1)*T(n-1,k-1) + (k+1)*T(n-1,k), with T(n,1) = T(n,n) = 1, 1 <= k <= n, read by rows.
%C Row sums are s(n) = 1, 2, 8, 53, 473, 5198, 67568, 1013513, 17229713, 327364538, ...
%H G. C. Greubel, <a href="/A156139/b156139.txt">Rows n=1..25 of triangle, flattened</a>
%H Leonard M. Smiley, <a href="http://www.math.uaa.alaska.edu/~smiley/BSfront.html">Completion of a Rational Function Sequence of Carlitz</a>, page 2.
%F Row sums s(n) = Sum_{k=1..n} T(n,k) seem to obey (n-2)*s(n) - (1-4*n+2*n^2)*s(n-1) + (3-5*n+2*n^2) * s(n-2)=0, n > 0. - _R. J. Mathar_, Jun 24 2011
%e Triangle begins with:
%e 1;
%e 1, 1;
%e 1, 6, 1;
%e 1, 23, 28, 1;
%e 1, 76, 250, 145, 1;
%e 1, 237, 1608, 2475, 876, 1;
%e 1, 722, 8802, 26847, 25056, 6139, 1;
%e 1, 2179, 43872, 231057, 418806, 268477, 49120, 1;
%e 1, 6552, 205994, 1725621, 5285520, 6486205, 3077730, 442089, 1;
%p A156139 := proc(n,k) option remember; if k= 1 or k=n then 1; else (2*n-k-1)*procname(n-1,k-1)+(k+1)*procname(n-1,k) ; end if; end proc:
%p seq(seq(A156139(n,k),k=1..n),n=1..10) ; # _R. J. Mathar_, Jun 24 2011
%t T[n_, 1]:= 1; T[n_, n_] := 1;
%t T[n_, k_]:= (2*n-k-1)*T[n-1, k-1] + (k+1)*T[n-1, k];
%t TableForm[Table[T[n, k], {n, 10}, {k, n}], TableAlignments -> Right];
%t Table[Table[T[n, k], {k, n}], {n, 10}]//Flatten
%o (PARI) {T(n, k) = if(k==1, 1, if(k==n, 1, (2*n-k-1)*T(n-1, k-1) + (k+1)*T(n-1, k)))};
%o for(n=1, 10, for(k=1, n, print1(T(n, k), ", "))) \\ _G. C. Greubel_, Feb 25 2019
%K nonn,tabl,easy
%O 1,5
%A _Roger L. Bagula_, Feb 04 2009