A386393 Triangle T(n, k) (1 <= k <= n) read by rows: T(n, k) is the numerator of R(n, k) where R(n, k) = R(n, k-1)/2 + R(n-1, k-1) for 1 < k <= n with R(n, 1) = R(n-1, n-1) for n > 1, R(1, 1) = 1.

1, 1, 3, 3, 7, 19, 19, 43, 99, 251, 251, 555, 1243, 2827, 6843, 6843, 14875, 32635, 72411, 162875, 381851, 381851, 819803, 1771803, 3860443, 8494747, 18918747, 43357211, 43357211, 92234139, 197168923, 423959707, 918096411, 2005424027, 4427023643, 9976746651

Offset

1,3

Comments

Denominator of R(n, k) is 2^((n-1)*(n-2)/2+k-1).

Links

Table of n, a(n) for n=1..36.

Max Alekseyev, Recursion for solution of f'(x) = exp(p*x)*f(q*x), answer to question on MathOverflow, 2025.

Example

Triangle begins:
       1;
       1,      3;
       3,      7,      19;
      19,     43,      99,     251;
     251,    555,    1243,    2827,    6843;
    6843,  14875,   32635,   72411,  162875,   381851;
  381851, 819803, 1771803, 3860443, 8494747, 18918747, 43357211;

Prog

(PARI) rows(n) = {my(A, v1 = vector(n, i, 0), v2 = vector(n, i, 0), v3); v1[1] = 1; v2[1] = [1]; for(i=2, n, v3 = v1; v1[1] = v3[i-1]; for(j=2, i, v1[j] = v1[j-1]/2 + v3[j-1]); A = 2^((i-1)*(i-2)/2); v2[i] = vector(i, j, A*2^(j-1)*v1[j])); v2}

Crossrefs

Cf. A305562.

Sequence in context: A036574 A174034 A081486 * A097334 A214496 A046631

Adjacent sequences: A386390 A386391 A386392 * A386394 A386395 A386396

Keyword

nonn,tabl,frac

Author

Mikhail Kurkov, Jul 20 2025

Status

approved