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

0,4

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.

Links

Table of n, a(n) for n=0..73.

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)

Example

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;

Prog

(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)

Crossrefs

Cf. A000120, A341392, A329369.

Sequence in context: A164848 A213514 A130827 * A070309 A287556 A353748

Adjacent sequences: A379814 A379815 A379816 * A379818 A379819 A379820

Keyword

nonn,base,tabf

Author

Mikhail Kurkov, Jan 03 2025

Status

approved