A380971 Irregular triangle T(n, k), n >= 0, k > 0, read by rows with row polynomials R(n, x) such that R(2n+1, x) = x*R(n, x) for n >= 0, R(2n,x) = wt(n)*x*((x+1)^wt(n) - x^wt(n)) + Sum_{k=1..wt(n)} k*x^k*T(n,k) for n > 0 with R(0,x) = 0 where wt(n) = A000120(n).

1, 2, 0, 1, 2, 4, 3, 0, 2, 2, 6, 0, 0, 1, 4, 12, 0, 2, 4, 3, 9, 9, 4, 0, 3, 2, 8, 0, 0, 2, 4, 16, 0, 2, 6, 3, 9, 12, 0, 0, 0, 1, 6, 28, 0, 4, 12, 3, 13, 21, 0, 0, 2, 4, 6, 27, 36, 0, 3, 9, 9, 4, 16, 24, 16, 5, 0, 4, 2, 10, 0, 0, 3, 4, 20, 0, 2, 8, 3, 9, 15, 0

Offset: 0

Author

Mikhail Kurkov, Feb 10 2025

Keywords

• nonn
• base
• tabf

Comments

Row n length is A000120(n) except for n = 2^k - 1 which are empty rows.

Examples

			Irregular triangle begins:
  -
  -
  1;
  -
  2;
  0,  1;
  2,  4;
  -
  3;
  0,  2;
  2,  6;
  0,  0, 1;
  4, 12;
  0,  2, 4;
  3,  9, 9;
  -
		

Crossrefs

Cf. A000120, A373183, A380179, A380944.

Programs

PARI

row(n) = if(n==0, [], my(x = 'x, A = 0, B = 0); forstep(i=logint(n, 2), 0, -1, A = if(bittest(n, i), B++; x*A, B*x*((x+1)^B - x^B) + sum(k=1, B, k*x^k*polcoeff(A, k, x)))); Vecrev(A/x))

Formula

Conjectures: (Start)

b(2^m*(2k+1)) = b(2^m*(2^wt(k)-1)) + Sum_{i=1..wt(k)} (i+1)^m*T(k,i)*(-1)^(wt(k)-i) for m >= 0, k >= 0 where b(n) = A380944(n) and where wt(n) = A000120(n). Note that this formula is recursive for k != 2^q - 1. We can also use b(2^m*(2^n-1)) = (n+1)^m - n*n!*c(m,n+1) for n >= 0, m >= 0 where c(n,k) = Sum_{i=0..n-k} Stirling2(k+i,k) for n >= 0, k >= 0.

A380179(n,k) = Sum_{i=0..2^(n+1)-1, [wt(i)<=(k+1)]*T(i,k+1)*(-1)^(wt(i)-k+1) for 0 <= k < n. (End)