A139382 -id:A139382 - cp's OEIS frontend

A342186 Triangle read by rows, matrix inverse of A139382.

1, -1, 1, 3, -4, 1, -21, 31, -11, 1, 315, -486, 196, -26, 1, -9765, 15381, -6562, 1002, -57, 1, 615195, -978768, 428787, -69688, 4593, -120, 1, -78129765, 124918731, -55434717, 9279163, -652999, 19833, -247, 1

Offset: 1

John Keith, Mar 04 2021

This triangle appears to be the q-analog of A008275 (Stirling numbers of the first kind) for q=2. However, A333142 has a similar definition.
Row sums of unsigned triangle are A006125 with offset 1.
|T(n,k)| is the number of descent free digraphs on [n] containing exactly k source nodes.  A descent in a digraph is a pair of vertices s->t such that s>t.  A descent free digraph is necessarily acyclic.  A source in an acyclic digraph is a node with indegree 0. - Geoffrey Critzer, Mar 05 2025

			The triangle begins:
           1;
          -1,         1;
           3,        -4,         1;
         -21,        31,       -11,       1;
         315,      -486,       196,     -26,       1;
       -9765,     15381,     -6562,    1002,     -57,     1;
      615195,   -978768,    428787,  -69688,    4593,  -120,    1;
   -78129765, 124918731, -55434717, 9279163, -652999, 19833, -247, 1;
  ...

John Keith, Rows n = 1..20 of triangle, flattened
Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, Counting acyclic and strong digraphs by descents, arXiv:1909.01550 [math.CO], 20 Mar 2020.

Cf. A008275, A139382, A333142, A333143, A006125 (row sums).

Columns of unsigned triangle: A005329, A203011, A000295, A203242.

A342186 := proc(n, k) if n = 1 and k = 1 then 1 elif k > n or k < 1 then 0 else
A342186(n-1, k-1) - (2^(n-1) - 1) * A342186(n-1, k) fi end:
for n from 1 to 8 do seq(A342186(n, k), k = 1..n) od; # Peter Luschny, Jun 28 2022

T[1, 1] := 1; T[n_, k_] := T[n, k] = If[k > n || k < 1, 0, T[n - 1, k - 1] - (2^(n - 1) - 1)*T[n - 1, k]]; Table[T[n, k], {n, 1, 8}, {k, 1, n}] (* after G. C. Greubel's program for A139382 *)

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) = 1/mat(nn); \\ Michel Marcus, Mar 18 2021

T(n,k) = T(n-1,k-1) - (2^(n-1)-1) * T(n-1,k), n, k >= 1, T(1, 1) = 1, T(n, 0) = 0.
For unsigned triangle, T(n, 1) = A005329(n-1); T(n, 2) = A203011(n-1); T(n, n-1) = A000295(n+1); T(n, n-2) = A203242(n-1).
T(n,k) = Sum_{j=k..n} (-1)^(n-j)*2^binomial(n-j,2)*qBinomial(n,j,2)*binomial(j,k), where qBinomial(n,k,2) is A022166(n,k). - Fabian Pereyra, Feb 08 2024

A022166 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 35, 15, 1, 1, 31, 155, 155, 31, 1, 1, 63, 651, 1395, 651, 63, 1, 1, 127, 2667, 11811, 11811, 2667, 127, 1, 1, 255, 10795, 97155, 200787, 97155, 10795, 255, 1, 1, 511, 43435, 788035, 3309747, 3309747, 788035, 43435, 511, 1

Offset: 0

N. J. A. Sloane

Also number of distinct binary linear [n,k] codes.
Row sums give A006116.
Central terms are A006098.
T(n,k) is the number of subgroups of the Abelian group (C_2)^n that have order 2^k. - Geoffrey Critzer, Mar 28 2016
T(n,k) is the number of k-subspaces of the finite vector space GF(2)^n. - Jianing Song, Jan 31 2020

			Triangle begins:
  1;
  1,   1;
  1,   3,    1;
  1,   7,    7,     1;
  1,  15,   35,    15,     1;
  1,  31,  155,   155,    31,    1;
  1,  63,  651,  1395,   651,   63,   1;
  1, 127, 2667, 11811, 11811, 2667, 127, 1;

J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 698.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

T. D. Noe, Rows n=0..50 of triangle, flattened
Octavio A. Agustín-Aquino, Archimedes' quadrature of the parabola and minimal covers, arXiv:1602.05279 [math.CO], 2016.
J. A. de Azcarraga and J. A. Macfarlane, Group Theoretical Foundations of Fractional Supersymmetry arXiv:hep-th/9506177, 1995.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
R. Mestrovic, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv:1409.3820 [math.NT], 2014.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
D. Slepian, A class of binary signaling alphabets, Bell System Tech. J. 35 (1956), 203-234.
D. Slepian, Some further theory of group codes, Bell System Tech. J. 39 1960 1219-1252.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
Eric W. Weisstein, q-Binomial Coefficient.
Wikipedia, q-binomial
Index entries for sequences related to binary linear codes
Index entries for sequences related to Gaussian binomial coefficients

Cf. A006516, A218449, A135950 (matrix inverse), A000225 (k=1), A006095 (k=2), A006096 (k=3), A139382.
Cf. this sequence (q=2), A022167 (q=3), A022168 (q=4), A022169 (q=5), A022170 (q=6), A022171 (q=7), A022172 (q=8), A022173 (q=9), A022174 (q=10), A022175 (q=11), A022176 (q=12), A022177 (q=13), A022178 (q=14), A022179 (q=15), A022180 (q=16), A022181 (q=17), A022182 (q=18), A022183 (q=19), A022184 (q=20), A022185 (q=21), A022186 (q=22), A022187 (q=23), A022188 (q=24).
Analogous triangles for other q: A015109 (q=-2), A015110 (q=-3), A015112 (q=-4), A015113 (q=-5), A015116 (q=-6), A015117 (q=-7), A015118 (q=-8), A015121 (q=-9), A015123 (q=-10), A015124 (q=-11), A015125 (q=-12), A015129 (q=-13), A015132 (q=-14), A015133 (q=-15).

q:=2; [[k le 0 select 1 else (&*[(1-q^(n-j))/(1-q^(j+1)): j in [0..(k-1)]]): k in [0..n]]: n in [0..20]]; // G. C. Greubel, Nov 17 2018

A005329 := proc(n)
   mul( 2^i-1,i=1..n) ;
end proc:
A022166 := proc(n,m)
   A005329(n)/A005329(n-m)/A005329(m) ;
end proc: # R. J. Mathar, Nov 14 2011

Table[QBinomial[n, k, 2], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 08 2016 *)
(* S stands for qStirling2 *) S[n_, k_, q_] /; 1 <= k <= n := S[n - 1, k - 1, q] + Sum[q^j, {j, 0, k - 1}]*S[n - 1, k, q]; S[n_, 0, ] := KroneckerDelta[n, 0]; S[0, k, ] := KroneckerDelta[0, k]; S[, , ] = 0;
T[n_, k_] /; n >= k := Sum[Binomial[n, j]*S[n - j, n - k, q]*(q - 1)^(k - j) /. q -> 2, {j, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 08 2020, after Vladimir Kruchinin *)

T(n,k)=polcoeff(x^k/prod(j=0,k,1-2^j*x+x*O(x^n)),n) \\ Paul D. Hanna, Oct 28 2006

qp = matpascal(9,2);
for(n=1,#qp,for(k=1,n,print1(qp[n,k],", "))) \\ Gerald McGarvey, Dec 05 2009

{q=2; T(n,k) = if(k==0,1, if (k==n, 1, if (k<0 || nG. C. Greubel, May 27 2018

def T(n,k): return gaussian_binomial(n,k).subs(q=2) # Ralf Stephan, Mar 02 2014

G.f.: A(x,y) = Sum_{k>=0} y^k/Product_{j=0..k} (1 - 2^j*x). - Paul D. Hanna, Oct 28 2006
For k = 1,2,3,... the expansion of exp( Sum_{n >= 1} (2^(k*n) - 1)/(2^n - 1)*x^n/n ) gives the o.g.f. for the k-th diagonal of the triangle (k = 1 corresponds to the main diagonal). - Peter Bala, Apr 07 2015
T(n,k) = T(n-1,k-1) + q^k * T(n-1,k). - Peter A. Lawrence, Jul 13 2017
T(m+n,k) = Sum_{i=0..k} q^((k-i)*(m-i)) * T(m,i) * T(n,k-i), q=2 (see the Sved link, page 337). - Werner Schulte, Apr 09 2019
T(n,k) = Sum_{j=0..k} qStirling2(n-j,n-k)*C(n,j) where qStirling2(n,k) is A139382. - Vladimir Kruchinin, Mar 04 2020

A005321 Upper triangular n X n (0,1)-matrices with no zero rows or columns.

Original entry on oeis.org

1, 1, 2, 10, 122, 3346, 196082, 23869210, 5939193962, 2992674197026, 3037348468846562, 6189980791404487210, 25285903982959247885402, 206838285372171652078912306, 3386147595754801373061066905042, 110909859519858523995273393471390010

Offset: 0

N. J. A. Sloane

T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..80
E. Andresen and K. Kjeldsen, On certain subgraphs of a complete transitively directed graph, Discrete Math. 14 (1976), no. 2, 103-119.
William T. Dugan, On the f-vectors of flow polytopes for the complete graph, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 101. See p. 3.
T. Lockman Greenough, Representation and enumeration of interval orders and semiorders, Ph.D. Thesis, Dartmouth, 1976.
T. L. Greenough, Enumeration of interval orders without duplicated holdings, Preprint, circa 1976. [Annotated scanned copy]
T. Lockman Greenough, Enumeration of interval orders without duplicated holdings, in Notices of the AMS, February 1976, page A-314.
T. L. Greenough and K. P. Bogart, The Representation and Enumeration of Interval Orders, Preprint, circa 1976. [Annotated scanned copy]
Hsien-Kuei Hwang, Emma Yu Jin, and Michael J. Schlosser, Asymptotics and statistics on Fishburn Matrices: dimension distribution and a conjecture of Stoimenow, arXiv:2012.13570 [math.CO], 2020.
Vít Jelínek, Counting Self-Dual Interval Orders, arXiv:1106.2261 [math.CO], 2011. See Corollary 2.4.
Vít Jelínek, Counting general and self-dual interval orders, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, pp. 599-614. See Corollary 2.4.
J. Longyear, T. Trotter, N. J. A. Sloane, Correspondence
Index entries for sequences related to binary matrices

Cf. A022493, A138265, A289314, A289315.

Column sums of A137252.

max = 14; f[x_] := Sum[ x^n*Product[ (2^i-1) / (1+(2^i-1)*x), {i, 1, n}], {n, 0, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 23 2011, after Vladeta Jovovic *)

a(n) = 1 + sum(k=2, n, binomial(n,k)*sum(i=2, k, (-1)^i*prod(j=i+1, k, 2^j - 1))); \\ Michel Marcus, Oct 13 2019

a(n) = Sum_{k=0..n} binomial(n,k)*A005327(k+1).
G.f.: Sum_{n >= 0} x^n*Product_{i = 1..n} ((2^i-1)/(1 + (2^i-1)*x)). - Vladeta Jovovic, Mar 10 2008
From Peter Bala, Jul 06 2017: (Start)
Two conjectural continued fractions for the o.g.f.:
1/(1 - x/(1 - x/(1 - 6*x/(1 - 9*x/(1 - 28*x/(1 - 49*x/(1 - ... - 2^(n-1)*(2^n - 1)*x/(1 - (2^n - 1)^2*x/(1 - ...)))))))));
1 + x/(1 - 2*x/(1 - 3*x/(1 - 12*x/(1 - 21*x/(1 - ... - 2^n*(2^n - 1)*x/(1 - (2^(n+1) - 1)*(2^n - 1)*x/(1 - ...))))))). Cf. A289314 and A289315. (End)
a(n) = (-1)^n*Sum_{k=0..n} qS2(n,k)*[k]!*(-1)^k, where qS2(n,k) is the triangle A139382, and [k]! is q-factorial, q=2. - Vladimir Kruchinin, Oct 10 2019
a(n) = 1 + Sum_{k=2..n} binomial(n,k)*Sum{i=2..k} (-1)^i*Product_{j=i+1..k} (2^j - 1). See Greenough. - Michel Marcus, Oct 13 2019

More terms from Max Alekseyev, Apr 27 2010

A135922 Inverse binomial transform of A006116, which is the sum of Gaussian binomial coefficients [n,k] for q=2.

Original entry on oeis.org

1, 1, 2, 6, 26, 158, 1330, 15414, 245578, 5382862, 162700898, 6801417318, 394502066810, 31849226170622, 3589334331706258, 566102993389615254, 125225331231990004138, 38920655753545108286254, 17021548688670112527781058, 10486973328106497739526535366

Offset: 0

Paul D. Hanna, Dec 06 2007

nonn

Let B = {v_1,v_2,...,v_n} be a basis for F_2^n.  a(n) is the number of subspaces of F_2^n that do not contain any of the vectors in B.  Moreover, for 1<=k<=n, let S be a size k subset of B. a(k) is the number of subspaces of F_2^n that do not contain any of the vectors in S but do contain all the vectors in B - S. - Geoffrey Critzer, May 03 2025
Also number of Stanley graphs on n nodes. For precise definition see Knuth (1997). - Alois P. Heinz, Sep 24 2019
Also the number of naturally labeled posets on [n] with height at most two. - David Bevan, Jul 28 2022; Nov 16 2023
Also the number of sign mappings X:([n] choose 2) -> {+,-} such that for any ordered 3-tuple aManfred Scheucher, Jan 05 2024

			O.g.f.: A(x) = 1 + x/(1-x) + x^2/((1-x)*(1-3x)) + x^3/((1-x)*(1-3x)*(1-7x)) + x^4/((1-x)*(1-3x)*(1-7x)*(1-15x)) + ...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 318.

Alois P. Heinz, Table of n, a(n) for n = 0..115
David Bevan, Gi-Sang Cheon and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.
Lucas Gagnon, The combinatorics of normal subgroups in the unipotent upper triangular group, arXiv:2012.00108 [math.CO], 2020.
D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission]
Zvi Rosen and Yan X. Zhang, Convex Neural Codes in Dimension 1, arXiv:1702.06907 [math.CO], 2017. Mentions this sequence.
R. P. Stanley, Problem 10572, The American Mathematical Monthly, 104(2) (1997), 168.
R. P. Stanley and S. C. Locke, Graphs without increasing paths: Solution to Problem 10572, The American Mathematical Monthly, 106(2) (1999), 168.

Cf. A006116, A323841, A323842, A323843, A139382, A006455, A356111.

b:= proc(n) option remember; add(mul(
      (2^(i+k)-1)/(2^i-1), i=1..n-k), k=0..n)
    end:
a:= proc(n) option remember;
      add(b(n-j)*binomial(n,j)*(-1)^j, j=0..n)
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Sep 24 2019

Table[SeriesCoefficient[Sum[x^n/Product[(1-(2^k-1)*x),{k,0,n}],{n,0,nn}],{x,0,nn}] ,{nn,0,20}] (* Vaclav Kotesovec, Aug 23 2013 *)
b[n_] := b[n] = Sum[Product[(2^(i+k)-1)/(2^i-1), {i, 1, n-k}], {k, 0, n}];
a[n_] := a[n] = Sum[(-1)^j b[n-j] Binomial[n, j], {j, 0, n}];
a /@ Range[0, 19] (* Jean-François Alcover, Mar 10 2020, after Alois P. Heinz *)

a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-(2^j-1)*x+x*O(x^n))), n)

# After Vladimir Kruchinin.
def a(n):
    @cached_function
    def T(n, k):
        if k == 1 or k == n: return 1
        return (2^k-1)*T(n-1, k) + T(n-1, k-1)
    return sum(T(n, k) for k in (1..n)) if n > 0 else 1
print([a(n) for n in (0..19)]) # Peter Luschny, Feb 26 2020

O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - (2^k-1)*x).
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-x*(2^k-1))/(1-x/(x-1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jan 16 2013
a(n) ~ c * 2^(n^2/4), where c = EllipticTheta[3,0,1/2]/QPochhammer[1/2,1/2] = 7.3719688014613... if n is even and c = EllipticTheta[2,0,1/2]/QPochhammer[1/2,1/2] = 7.3719494907662... if n is odd. - Vaclav Kotesovec, Aug 23 2013
a(n) = Sum_{k=0..n} qStirling2(n,k), where qStirling2 is the triangle A139382. - Vladimir Kruchinin, Feb 26 2020
G.f.: f(1), where f(y) = 1 + x*((y-1)*f(y) + f(2*y)). - David Bevan, Jul 28 2022
E.g.f.:  exp(-x)*g(x) where g(x) is the e.g.f. for A006116. (given in D. E. Knuth link) - Geoffrey Critzer, May 03 2025

References for Stanley graphs added by David Bevan, Jul 24 2024

A333143 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = 3, with 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 21, 18, 1, 1, 85, 255, 58, 1, 1, 341, 3400, 2575, 179, 1, 1, 1365, 44541, 106400, 24234, 543, 1, 1, 5461, 580398, 4300541, 3038714, 221886, 1636, 1, 1, 21845, 7550635, 172602038, 371984935, 83805218, 2010034, 4916, 1

Offset: 0

Peter Luschny, Mar 09 2020

			[0] 1
[1] 1, 1
[2] 1, 5,     1
[3] 1, 21,    18,      1
[4] 1, 85,    255,     58,        1
[5] 1, 341,   3400,    2575,      179,       1
[6] 1, 1365,  44541,   106400,    24234,     543,      1
[7] 1, 5461,  580398,  4300541,   3038714,   221886,   1636,    1
[8] 1, 21845, 7550635, 172602038, 371984935, 83805218, 2010034, 4916, 1

T(n, 1) = A002450(n), T(n, n-1) = A000340(n).

Cf. A139382 (q=2), A333142.

qStirling2 := proc(n, k, q) option remember; with(QDifferenceEquations):
if n = 0 then return 0^k fi; if k = 0 then return n^0 fi;
qStirling2(n-1, k-1, p) + QBrackets(k+1, p)*qStirling2(n-1, k, p);
subs(p = q, expand(%)) end:
seq(seq(qStirling2(n, k, 3), k=0..n), n=0..9);

qStirling2[n_, k_, q_] /; 1 <= k <= n := (* q^(k-1) *) qStirling2[n - 1, k - 1, q] + Sum[q^j, {j, 0, k - 1}] qStirling2[n - 1, k, q];
qStirling2[n_, 0, _] := KroneckerDelta[n, 0];
qStirling2[0, k_, _] := KroneckerDelta[0, k];
qStirling2[, , _] = 0;
Table[qStirling2[n + 1, k + 1, 3], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 11 2020 *)

qStirling2(n, k, q) = qStirling2(n-1, k-1, q) + qBrackets(k+1, q)*qStirling2(n-1, k, q) with boundary values 0^k if n = 0 and n^0 if k = 0.

Note that also a second definition is used in the literature which has an additional factor q^k attached to the first term in the equation above. The two versions differ by a factor of q^binomial(k,2).

A308326 The q-analog T(q; n,k) of the triangle A163626 for 0 <= k <= n, for q = 2.

Original entry on oeis.org

1, 1, -1, 1, -4, 3, 1, -13, 33, -21, 1, -40, 270, -546, 315, 1, -121, 2010, -10080, 17955, -9765, 1, -364, 14433, -165270, 707805, -1171800, 615195, 1, -1093, 102123, -2580081, 24421005, -95765355, 151953165, -78129765, 1, -3280, 718140, -39416076, 795752370, -6790268520, 25331269320, -39221142030, 19923090075

Offset: 0

Werner Schulte, May 23 2019

The formulas are given for the general case depending on some fixed integer q. The terms are valid if q = 2.

Special cases: T(0; n,k) = (-1)^k * binomial(n,k) for 0 <= k <= n and T(1; n,k) = A163626(n,k) for 0 <= k <= n.

			If q = 2 the triangle T(2; n,k) starts:
n\k:  0     1      2        3        4         5         6         7
====================================================================
  0:  1
  1:  1    -1
  2:  1    -4      3
  3:  1   -13     33      -21
  4:  1   -40    270     -546      315
  5:  1  -121   2010   -10080    17955     -9765
  6:  1  -364  14433  -165270   707805  -1171800    615195
  7:  1 -1093 102123 -2580081 24421005 -95765355 151953165 -78129765
etc.

Cf. A000007, A007318, A022166, A022167, A139382, A163626.

q = 2; {T(n,k) = if(k<0 || k>n, 0, if(k==0, 1, if(q==1, (k+1) * T(n-1,k) - k * T(n-1,k-1), ((q^(k+1) - 1)/(q - 1)) * T(n-1,k) - ((q^k - 1)/(q - 1)) * T(n-1,k-1))))};
for(n=0, 9, for(k=0, n, print1(T(n,k), ", "))) \\ Werner Schulte, May 26 2019

T(q; n,k) = [k+1]_q * T(q; n-1,k) - [k]_q * T(q; n-1,k-1) for 1 <= k <= n with initial values T(q; n,0) = 1 for n >= 0 and T(q; i,j) = 0 if i < j or j < 0 where [i]_q = (q^i - 1)/(q - 1) for i >= 0.
T(q; n,k) = (1/q^binomial(k+1,2)) * (Sum_{j=0..k} (-1)^j * [k,j]_q * q^binomial(k-j,2) * ([j+1]_q)^n) for 0 <= k <= n and q not equal zero where [m,i]_q are the q-binomials (here A022166 for q = 2) and [i]_q = (q^i - 1)/(q - 1) for i >= 0.
Sum_{k=0..n} T(q; n,k) = A000007(n) for n >= 0.
T(q; n,k)/T(q; k,k) give the q-analogs of the Stirling numbers of the second kind (for q = 2 see A139382, but offset 1).
T(q; n,n) = (-1)^n * Product_{j=1..n} [j]_q for n>=0 with empty product 1 (case n = 0) where [i]_q = (q^i - 1)/(q - 1) for i >= 0.
T(q; n,1) = -[n,1]_(q+1) for n >= 1 where [m,i]_q are the q-binomials (here A022166 for q = 2 and A022167 for q = 3).
G.f. of column k: col(q; t,k) = Sum_{n>=k} T(q; n,k)*t^n = ((-t)^k/(1-t)) * Product_{j=1..k} ([i]_q/(1-[i+1]_q*t)) for k>=0 with empty product 1 (case k=0) and [i]_q = i if q = 1 otherwise (q^i-1)/(q-1) for i>=0.

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.

Original entry on oeis.org

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

A342186 Triangle read by rows, matrix inverse of A139382.

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A022166 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2.

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A005321 Upper triangular n X n (0,1)-matrices with no zero rows or columns.

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A135922 Inverse binomial transform of A006116, which is the sum of Gaussian binomial coefficients [n,k] for q=2.

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A333143 Triangle read by rows: T(n, k) = qStirling2(n, k, q) for q = 3, with 0 <= k <= n.

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A308326 The q-analog T(q; n,k) of the triangle A163626 for 0 <= k <= n, for q = 2.

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