A376586 Irregular table T(n, k), n >= 0, k >= 0, read by rows with row polynomials P(n-1, n-1) for n > 0 with T(0, 0) = 1 where P(n, k) = (-1)^(n-k)*z^f(n-k+1)*P(n, k-1) + P(k-1, k-1) for 0 < k <= n with P(n, k) = 0 for k > n, P(n, 0) = 1 for n >= 0 and where f(2n+1) = n, f(2n) = 2n for n >= 0.

1, 1, 2, 3, 0, -1, 5, 0, -2, -1, 8, 0, -5, -2, 0, 0, 0, 1, 13, 0, -10, -5, 1, 0, 0, 2, 0, 1, 21, 0, -20, -10, 3, 2, 0, 5, 0, 2, 0, 0, 0, 0, 0, -1, 34, 0, -38, -20, 9, 6, 1, 10, 0, 3, 0, 0, 0, 0, 0, -2, 0, 0, -1, 55, 0, -71, -38, 22, 18, 2, 20, 0, 4, -2, -2, 0, 0

Offset: 0

Author

Mikhail Kurkov, Sep 29 2024

Keywords

• sign
• tabf

Comments

Row n length is A131355(n).

Conjecture: antidiagonal sums equal A000041.

Examples

			Irregular table begins:
   1;
   1;
   2;
   3, 0,  -1;
   5, 0,  -2,  -1;
   8, 0,  -5,  -2, 0, 0, 0,  1;
  13, 0, -10,  -5, 1, 0, 0,  2, 0, 1;
  21, 0, -20, -10, 3, 2, 0,  5, 0, 2, 0, 0, 0, 0, 0, -1;
  34, 0, -38, -20, 9, 6, 1, 10, 0, 3, 0, 0, 0, 0, 0, -2, 0, 0, -1;
		

Crossrefs

Cf. A000041, A000045, A084610, A131355.

Programs

PARI

f(n) = if(n%2, (n-1)/2, n)
rows_upto(n) = my(v1); v1 = vector(n+1, i, 1); for(i=2, n, for(j=i+1, n+1, v1[j] = v1[i] + (-1)^(j-i+1)*z^f(j-i)*v1[j])); v1 = vector(n+1, i, Vecrev(v1[i]))

Formula

Conjectures: (Start)

T(n, 0) = A000045(n+1) for n >= 0.

T(n, 1) = 0 for n >= 0.

Sum_{j=0..2*(floor(k/2)+1)} A084610(floor(k/2)+1, j)*T(n+j, k) = 0 for n >= b(k), k >= 0 where b(k) is some nonnegative integer sequence (with a single exception at k = 1).

G.f. for k-th column is Q_k(x)/(1-x-x^2)^(floor(k/2)+1) for k >= 0 where Q_k(x) is some family of polynomials (with a single exception at k = 1). (End)