A357438 Triangle T(n,k) read by rows, defined by the equation f(x, y) := Sum_{n, k} T(n, k) * y^k * x^n = 1/(1 - x*y - x^2*y*f(x, y+1)).

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 2, 6, 6, 1, 0, 5, 16, 20, 10, 1, 0, 15, 51, 71, 50, 15, 1, 0, 52, 186, 281, 231, 105, 21, 1, 0, 203, 759, 1223, 1114, 616, 196, 28, 1, 0, 877, 3409, 5795, 5701, 3564, 1428, 336, 36, 1, 0, 4140, 16655, 29634, 31011, 21187, 9780

Offset: 1

Author

Michael Somos, Sep 27 2022

Keywords

• nonn
• tabl

Comments

Row sums are A000110.

Examples

			Triangle starts:
  1,
  0,   1,
  0,   1,    1,
  0,   1,    3,    1,
  0,   2,    6,    6,    1,
  0,   5,   16,   20,   10,    1,
  0,  15,   51,   71,   50,   15,    1,
  0,  52,  186,  281,  231,  105,   21,  1,
  0, 203,  759, 1223, 1114,  616,  196,  28,  1,
  0, 877, 3409, 5795, 5701, 3564, 1428, 336, 36, 1,
  ...
		

Crossrefs

Cf. A000110, A049347, A074664.

Programs

Mathematica

T[ n_, k_] := If[n < 0, 0, Coefficient[SeriesCoefficient[ Nest[ 1/(1 - x*y - x^2*y*(#/.y -> y+1))&, 1 + O[x], Ceiling[n/2]], {x, 0, n}], y, k]];

PARI

{T(n, k) = if(n < 0, 0, f = 1 + O(x); forstep(i=1, n, 2, f = 1/(1 - x*y - x^2*y*subst(f, y, y+1))); polcoef(polcoef(f, n), k))};

Formula

f(x, -1) = 1/(1 + x + x^2).

x + x^2*f(x, 2) = 1 - 1/f(x, 1) is g.f. for A074664.