cp's OEIS Frontend

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.

%I A145140 #10 Oct 04 2018 20:17:58
%S A145140 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,
%T A145140 22,7,5,1,1,2,53,5989,1069,1207,7,23,1,1,3,611,2743,93791,149,1213,1,
%U A145140 13,1,1,4,101,25523,5419,20071,397,3253,1,29,1,1,5,32419,11017,30731,21757
%N 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.
%F A145140 See program.
%e A145140 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
%e A145140 As triangle:
%e A145140   1
%e A145140   0 1
%e A145140   0 1/2 1/2
%e A145140   0 1/3 1/2 1/6
%e A145140   1 1/4 11/24 1/4 1/24
%e A145140   1 6/5 5/12 7/24 1/12 1/120
%p A145140 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);
%t A145140 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_ *)
%Y A145140 Denominators of T(n, k): A145141. T(n, k)*(n-1)!: A145142.
%Y A145140 Row sums give: A003269, A017898(n+3).
%Y A145140 Cf. A145153, A000012, A001477, A000217, A000292, A145126, A145127, A145128, A145129, A145130.
%K A145140 frac,nonn,tabl
%O A145140 1,13
%A A145140 _Alois P. Heinz_, Oct 03 2008