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A145142 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.

%I A145142 #14 Oct 04 2018 20:16:08
%S A145142 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 A145142 85,15,1,5040,9960,6804,2464,735,175,21,1,80640,89040,71868,29932,
%U A145142 8449,1960,322,28,1,1088640,1231776,789984,375164,112644,25473,4536,546,36,1
%N A145142 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 A145142 Alois P. Heinz, <a href="/A145142/b145142.txt">Rows n = 1..45, flattened</a>
%F A145142 See program.
%e A145142 Triangle begins:
%e A145142     1;
%e A145142     0,   1;
%e A145142     0,   1,   1;
%e A145142     0,   2,   3,   1;
%e A145142    24,   6,  11,   6,   1;
%e A145142   120, 144,  50,  35,  10,  1;
%p A145142 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 A145142 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 A145142 T(n,k)/(n-1)! gives: A145140 / A145141.
%Y A145142 Columns 0-9 give: A052581, A145143, A145144, A145145, A145146, A145147, A145148, A145149, A145150.
%Y A145142 Diagonal and lower diagonals 1-3 give: A000012, A000217, A000914, A001303.
%Y A145142 Cf. A145153, A001477, A000292, A145126, A145127, A145128, A145129, A145130.
%Y A145142 Row sums are in A052593.
%K A145142 nonn,tabl
%O A145142 1,8
%A A145142 _Alois P. Heinz_, Oct 03 2008