A145140 Numerators of triangle T(n,k), n>=1, 0<=k<=n - 1, read by rows: T(n,k) 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, 1, 1, 1, 1, 1, 11, 1, 1, 1, 6, 5, 7, 1, 1, 1, 5, 317, 5, 17, 1, 1, 1, 83, 27, 22, 7, 5, 1, 1, 2, 53, 5989, 1069, 1207, 7, 23, 1, 1, 3, 611, 2743, 93791, 149, 1213, 1, 13, 1, 1, 4, 101, 25523, 5419, 20071, 397, 3253, 1, 29, 1, 1, 5, 32419, 11017, 30731, 21757

Offset

1,13

Links

Table of n, a(n) for n=1..71.

Formula

See program.

Example

1, 0, 1, 0, 1/2, 1/2, 0, 1/3, 1/2, 1/6, 1, 1/4, 11/24, 1/4, 1/24, 1, 6/5, 5/12, 7/24, 1/12, 1/120, 1, 5/3, 317/360, 5/16, 17/144, 1/48, 1/720 ... = A145140/A145141
As triangle:
  1
  0 1
  0 1/2 1/2
  0 1/3 1/2 1/6
  1 1/4 11/24 1/4 1/24
  1 6/5 5/12 7/24 1/12 1/120

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)-> coeff(row(n)(x), x, k): seq(seq(numer(T(n, k)), k=0..n-1), n=1..14);

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] // Numerator, {n, 1, 14}] // Flatten (* Jean-François Alcover, Feb 04 2014, after Alois P. Heinz *)

Crossrefs

Denominators of T(n, k): A145141. T(n, k)*(n-1)!: A145142.

Row sums give: A003269, A017898(n+3).

Cf. A145153, A000012, A001477, A000217, A000292, A145126, A145127, A145128, A145129, A145130.

Sequence in context: A280443 A107045 A351449 * A010195 A010193 A214679

Adjacent sequences: A145137 A145138 A145139 * A145141 A145142 A145143

Keyword

frac,nonn,tabl

Author

Alois P. Heinz, Oct 03 2008

Status

approved