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%I #14 Oct 04 2018 20:16:08
%S 1,0,1,0,1,1,0,2,3,1,24,6,11,6,1,120,144,50,35,10,1,720,1200,634,225,
%T 85,15,1,5040,9960,6804,2464,735,175,21,1,80640,89040,71868,29932,
%U 8449,1960,322,28,1,1088640,1231776,789984,375164,112644,25473,4536,546,36,1
%N Triangle T(n,k), n>=1, 0<=k<=n-1, read by rows: T(n,k)/(n-1)! is the coefficient of x^k in polynomial p_n for the n-th row sequence of A145153.
%H Alois P. Heinz, <a href="/A145142/b145142.txt">Rows n = 1..45, flattened</a>
%F See program.
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 3, 1;
%e 24, 6, 11, 6, 1;
%e 120, 144, 50, 35, 10, 1;
%p row:= proc(n) option remember; local f,i,x; f:= unapply(simplify(sum('cat(a||i) *x^i', 'i'=0..n-1) ), x); unapply(subs(solve({seq(f(i+1)= coeftayl(x/ (1-x-x^4)/ (1-x)^i, x=0, n), i=0..n-1)}, {seq(cat(a||i), i=0..n-1)}), sum('cat(a||i) *x^i', 'i'=0..n-1) ), x); end: T:= (n,k)-> `if`(k<0 or k>=n,0, coeff(row(n)(x),x,k)*(n-1)!): seq(seq(T(n,k), k=0..n-1), n=1..12);
%t row[n_] := Module[{f, eq}, f = Function[x, Sum[a[k]*x^k, {k, 0, n-1}]]; eq = Table[f[k+1] == SeriesCoefficient[x/(1-x-x^4)/(1-x)^k, {x, 0, n}], {k, 0, n-1}]; Table[a[k], {k, 0, n-1}] /. Solve[eq] // First]; Table[row[n]*(n-1)!, {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Feb 04 2014, after _Alois P. Heinz_ *)
%Y T(n,k)/(n-1)! gives: A145140 / A145141.
%Y Columns 0-9 give: A052581, A145143, A145144, A145145, A145146, A145147, A145148, A145149, A145150.
%Y Diagonal and lower diagonals 1-3 give: A000012, A000217, A000914, A001303.
%Y Cf. A145153, A001477, A000292, A145126, A145127, A145128, A145129, A145130.
%Y Row sums are in A052593.
%K nonn,tabl
%O 1,8
%A _Alois P. Heinz_, Oct 03 2008
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