A383019 Triangle T(n,k) read by rows (n >= 0, k >= 0) with g.f. 1/(1 - f(0)*x - x*y/(1 - f(1)*x - x*y/(1 - f(2)*x - x*y/(1 - f(3)*x - x*y/(1 - f(4)*x - x*y/(1 - ...)))))) where f(n) = n + 1 for n >= 0.

1, 1, 1, 1, 4, 2, 1, 11, 16, 5, 1, 26, 80, 64, 14, 1, 57, 324, 490, 256, 42, 1, 120, 1170, 2944, 2730, 1024, 132, 1, 247, 3948, 15403, 22400, 14322, 4096, 429, 1, 502, 12776, 73960, 157564, 152064, 72072, 16384, 1430, 1, 1013, 40264, 335856, 1004400, 1368796, 953344, 351780, 65536, 4862

Offset: 0

Author

Mikhail Kurkov, Apr 12 2025

Keywords

• nonn
• tabl

Comments

Conjecture: T(n,k) is the number of paths of length n + k steps from (0,0) to (2*n,0) on or above the x-axis with steps U = (1,1), D = (1,-1), and L = (2,0), where the level steps L at height m have f(m) = m + 1 colors for m >= 0. The same seems to work for any f(n). Also, it looks like that if we modify g.f. to 1/(1 - f(0)*x - g(0)*x*y/(1 - f(1)*x - g(1)*x*y/(1 - f(2)*x - g(2)*x*y/(1 - f(3)*x - g(3)*x*y/(1 - f(4)*x - g(4)*x*y/(1 - ...)))))), then steps U at height m have g(m) colors for m >= 0.

Examples

			Triangle begins:
  1;
  1,    1;
  1,    4,     2;
  1,   11,    16,     5;
  1,   26,    80,    64,     14;
  1,   57,   324,   490,    256,     42;
  1,  120,  1170,  2944,   2730,   1024,   132;
  1,  247,  3948, 15403,  22400,  14322,  4096,   429;
  1,  502, 12776, 73960, 157564, 152064, 72072, 16384, 1430;
  ...
		

Crossrefs

Cf. A302285 (row sums).

Programs

PARI

rows_upto(n) = my(A = 1, x = 'x, y = 'y, v1); forstep(j=n, 1, -1, A = 1 - j*x - x*y/A + x*O(x^n)); v1 = Vec(1/A); v1 = vector(n+1, i, Vecrev(v1[i]))

Formula

Conjecture: row polynomials are R(n,0,x) where R(n,k,x) = R(n-1,k+1,x) + x*Sum_{i=0..n-1} Sum_{j=0..k} binomial(k,j)*R(n-i-1,j,x)*R(i,k-j,x) for n > 0, k >= 0 with R(0,k,x) = 1 for k >= 0.

Conjecture: using modified g.f. given in the comments, the n-th row polynomial is v_n for n > 0 where we start with vector v of fixed length m with elements v_i = Sum_{j=0..i-1} f(j) + g(j)*x, reserve t as an empty vector of fixed length m and for i=1..m-1, for j=i+1..m, apply t := v (at the beginning of each cycle for i) and also apply v_j := [j = (i+1)]*f(0)*v_{j-1} + [j > (i+1)]*(v_{j-1} + f(j-i-1)*t_{j-1}) + g(j-i-1)*x*v_j.