A379817 Irregular table T(n, k), n >= 0, k >= 0, read by rows such that T(n,k) = f(n,k)/f(2^k-1,k) where f(n,k) is defined in Comments.

1, 1, 1, 3, 2, 1, 3, 1, 7, 4, 3, 7, 2, 7, 12, 3, 1, 7, 6, 1, 15, 8, 7, 15, 4, 17, 26, 6, 3, 17, 13, 2, 31, 42, 9, 7, 31, 21, 3, 15, 50, 30, 4, 1, 15, 25, 10, 1, 31, 16, 15, 31, 8, 37, 54, 12, 7, 37, 27, 4, 69, 88, 18, 17, 69, 44, 6, 37, 112, 63, 8, 3, 37, 56

Offset: 0

Author

Mikhail Kurkov, Jan 03 2025

Keywords

• nonn
• base
• tabf

Comments

Here f(n,k) = b(2^k*(2n+1)) - Sum_{j=1..k} b(2^(j-1)*(2n+1))*R(k,j) for n >= 0, k >= 0 where b(n) = A329369(n) and where R(k,j) is the unique solution to b(2^k*(2^i-1)) = Sum_{j=1..k} b(2^(j-1)*(2^i-1))*R(k,j) for k > 0, 1 <= i <= k.

Row n length is A000120(n) + 1.

Examples

			Irregular table begins:
   1;
   1,  1;
   3,  2;
   1,  3,  1;
   7,  4;
   3,  7,  2;
   7, 12,  3;
   1,  7,  6,  1;
  15,  8;
   7, 15,  4;
  17, 26,  6;
   3, 17, 13,  2;
  31, 42,  9;
   7, 31, 21,  3;
  15, 50, 30,  4;
   1, 15, 25, 10, 1;
		

Crossrefs

Cf. A000120, A341392, A329369.

Programs

PARI

upto(n) = my(A, v1); v1 = vector(n+1, i, 0); v1[1] = 1; for(i=1, n, v1[i+1] = v1[i\2+1] + if(i%2, 0, A = 1 << valuation(i/2, 2); v1[i/2-A+1] + v1[i-A+1])); v1 \\ from A329369
R(k) = my(v1, M1, M2); v1 = upto(2^k*(2^k-1)); M1 = matrix(k, k, i, j, v1[2^(j-1)*(2^i-1)+1]); M2 = matrix(k, 1, i, j, v1[2^k*(2^i-1)+1]); M1 = matsolve(M1, M2)
row(n) = my(A = hammingweight(n), v1, v2, v3); v1 = upto(2^A*(2*n+1)); v2 = vector(A, i, R(i)); v3 = vector(A, i, (v1[2^i*(2*n+1)+1] - sum(j=1, i, v1[2^(j-1)*(2*n+1)+1]*v2[i][j,1]))/(v1[2^i*(2*(2^i-1)+1)+1] - sum(j=1, i, v1[2^(j-1)*(2*(2^i-1)+1)+1]*v2[i][j,1]))); concat(v1[n+1], v3)

Formula

Conjectures: (Start)

f(2^k-1,k) = ((k+1)!)^2 for k >= 0.

R(k,j) = -Stirling1(k+2, j+1) for k > 0, 1 <= j <= k.

T(2^n-1, k) = Stirling2(n+1, k+1) for n >= 0, 0 <= k <= n.

T(n,k) = c(n,wt(n)-k) for n >= 0, 0 <= k <= wt(n) where c(2n+1,k) = c(n,k) + (wt(n)-k+2)*c(n,k-1), c(2n,k) = (wt(n)-k+1)*c(2n+1,k) for n > 0, k > 0 with c(n,0) = A341392(n) for n >= 0, c(0,k) = 0 for k > 0 and where wt(n) = A000120(n). (End)