A379819 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, 4, 2, 1, 3, 1, 10, 4, 4, 8, 2, 10, 13, 3, 1, 7, 6, 1, 22, 8, 10, 18, 4, 28, 30, 6, 4, 20, 14, 2, 49, 47, 9, 10, 36, 22, 3, 22, 56, 31, 4, 1, 15, 25, 10, 1, 46, 16, 22, 38, 8, 64, 64, 12, 10, 46, 30, 4, 118, 102, 18, 28, 88, 48, 6, 64, 138, 68, 8, 4

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) = A379818(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..71.

Formula

Conjectures: (Start)

f(2^k-1,k) = (k+1)*A130032(k+1) for k >= 0.

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) + [(wt(n)-k+1) > 0]*c(n,k-1) 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;
   4,  2;
   1,  3,  1;
  10,  4;
   4,  8,  2;
  10, 13,  3;
   1,  7,  6,  1;
  22,  8;
  10, 18,  4;
  28, 30,  6;
   4, 20, 14,  2;
  49, 47,  9;
  10, 36, 22,  3;
  22, 56, 31,  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[i\(4*A)+1])); v1 \\ from A379818
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, A379818.

Sequence in context: A309975 A010124 A271310 * A071406 A337063 A375119

Adjacent sequences: A379816 A379817 A379818 * A379820 A379821 A379822

Keyword

nonn,base,tabf

Author

Mikhail Kurkov, Jan 03 2025

Status

approved