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.

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, 85, 15, 1, 5040, 9960, 6804, 2464, 735, 175, 21, 1, 80640, 89040, 71868, 29932, 8449, 1960, 322, 28, 1, 1088640, 1231776, 789984, 375164, 112644, 25473, 4536, 546, 36, 1

Offset: 1

Author

Alois P. Heinz, Oct 03 2008

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    1;
    0,   1;
    0,   1,   1;
    0,   2,   3,   1;
   24,   6,  11,   6,   1;
  120, 144,  50,  35,  10,  1;
		

Links

• Alois P. Heinz, Rows n = 1..45, flattened

Crossrefs

T(n,k)/(n-1)! gives: A145140 / A145141.

Columns 0-9 give: A052581, A145143, A145144, A145145, A145146, A145147, A145148, A145149, A145150.

Diagonal and lower diagonals 1-3 give: A000012, A000217, A000914, A001303.

Cf. A145153, A001477, A000292, A145126, A145127, A145128, A145129, A145130.

Row sums are in A052593.

Programs

Maple

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);

Mathematica

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 *)

Formula

See program.