A342186 -id:A342186 - cp's OEIS frontend

A355282 Triangle read by rows: T(n, k) = Sum_{i=1..n-k} qStirling1(n-k, i) * qStirling2(n-1+i, n-1) for 0 < k < n with initial values T(n, 0) = 0^n and T(n, n) = 1 for n >= 0, here q = 2.

1, 0, 1, 0, 1, 1, 0, 9, 4, 1, 0, 343, 79, 11, 1, 0, 50625, 6028, 454, 26, 1, 0, 28629151, 1741861, 68710, 2190, 57, 1, 0, 62523502209, 1926124954, 38986831, 656500, 9687, 120, 1, 0, 532875860165503, 8264638742599, 84816722571, 734873171, 5760757, 40929, 247, 1

Offset: 0

Werner Schulte, Jun 26 2022

We aim at a q-generalization of the Comtet-Lehmer numbers A354794, which are the case q = 1. Here we consider the case q = 2. The generalization is based on the qStirling numbers, for qStirling1 see A342186 and for qStirling2 see A139382. The general construction is as follows:
Let q <> 1 be a fixed integer and f_q(k) = (q^k - 1)/(q - 1) for k >= 0. Define triangle M(q; n, k) for 0 <= k <= n by M(q; n, 0) = 0^n for n >= 0, and M(q; n, k) = 0 for k > n, and M(q; n, k) = M(q; n-1, k-1) + M(q; n-1, k) * f_q(k) for 0 < k <= n. Then M(q; n, n) = 1 for n >= 0 and the matrix inverse I_q = M_q^(-1) exists. Next define the triangle T(q; n, k) for 0 <= k <= n by T(q; n, 0) = 0^n for n >= 0 and T(q; n, k) = Sum_{i=0..n-k} I(q; n-k, i) * M(q; n-1+i, n-1) for 0 < k <= n. Take account of lim_{q->1} (q^n - 1)/(q - 1) = n for n >= 0.
Conjecture: T(q; n+1, 1) = Sum_{i=0..n} I(q; n, i) * M(q; n+i, n) = (f_q(n))^n = ((q^n - 1)/(q - 1))^n for n >= 0.
Conjecture: T(q; n, k) = (Sum_{i=0..n-k} (-1)^i * q-binomial(n-1-i, k-1) * binomial(n-1, i) * q^((n-k)*(n-k-i))) / (q - 1)^(n-k) for 0 < k <= n.

			Triangle T(n, k) for 0 <= k <= n starts:
n\k : 0               1             2           3         4       5     6   7 8
===============================================================================
  0 : 1
  1 : 0               1
  2 : 0               1             1
  3 : 0               9             4           1
  4 : 0             343            79          11         1
  5 : 0           50625          6028         454        26       1
  6 : 0        28629151       1741861       68710      2190      57     1
  7 : 0     62523502209    1926124954    38986831    656500    9687   120   1
  8 : 0 532875860165503 8264638742599 84816722571 734873171 5760757 40929 247 1
  etc.

Cf. A022166, A139382, A342186, A354794, A055601 (column 1), A125128 (1st subdiagonal).

# using qStirling2 from A333143.
A355282 := proc(n, k) if k = 0 then 0^n elif n = k then 1 else
add(A342186(n - k, i)*qStirling2(n + i - 2, n - 2, 2), i = 1..n-k) fi end:
seq(print(seq(A355282(n, k), k = 0..n)), n = 0..8); # Peter Luschny, Jun 28 2022

mat(nn) = my(m = matrix(nn, nn)); for (n=1, nn, for(k=1, nn, m[n, k] = if (n==1, if (k==1, 1, 0), if (k==1, 1, (2^k-1)*m[n-1, k] + m[n-1, k-1])); ); ); m; \\ A139382
tabl(nn) = my(m=mat(3*nn), im=1/m); matrix(nn, nn, n, k, n--; k--; if (k==0, 0^n, kMichel Marcus, Jun 27 2022

Conjecture: T(n+1, 1) = (2^n - 1)^n for n >= 0.

Conjecture: T(n, k) = Sum_{i=0..n-k} (-1)^i * binomial(n-1, i) * [n-1-i, k-1]_2 * 2^((n-k)*(n-k-i)) for 0 < k <= n and T(n, 0) = 0^n for n >= 0, where [x, y]_2 = A022166(x, y).

cp's OEIS Frontend

A355282 Triangle read by rows: T(n, k) = Sum_{i=1..n-k} qStirling1(n-k, i) * qStirling2(n-1+i, n-1) for 0 < k < n with initial values T(n, 0) = 0^n and T(n, n) = 1 for n >= 0, here q = 2.

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