A022166 -id:A022166 - 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).

A359313 Triangular array read by rows. T(n,k) is the number of Green's H-classes contained in the D-class of rank k matrices in the semigroup Mat_n(F_2) of n X n matrices over the field F_2. n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 49, 49, 1, 1, 225, 1225, 225, 1, 1, 961, 24025, 24025, 961, 1, 1, 3969, 423801, 1946025, 423801, 3969, 1, 1, 16129, 7112889, 139499721, 139499721, 7112889, 16129, 1, 1, 65025, 116532025, 9439094025, 40315419369, 9439094025, 116532025, 65025, 1

Offset: 0

Geoffrey Critzer, Dec 25 2022

For all a,b in the semigroup Mat_n(F_2), aDb if and only if rank(a)=rank(b). Also, aHb if and only if the row(a)=row(b) and col(a)=col(b). So the H classes correspond to ordered pairs (U,W) of subspaces of F_2^n with dim(U) = dim(W). Let a in Mat_n(F_2) be such that col(a) = U and row(a)=W. The size of H_a, the H-class containing a is |GL_d(F_2)| where d=dim(U). H_a contains an idempotent if and only if col(a) + perp(row(a)) is a direct sum decomposition of F_2^n where perp(X)={v in F_2^n: v*x = 0 for all x in X}.

Let H_a,H_b be H-classes in Mat_n(F_2). Let H_a ~ H_b if and only if col(a) is contained in col(b) and row(a) is contained in row(b). Then ~ is a partial order relation on the set of all H-classes in Mat_n(F_q). The poset is isomorphic to a typical n-interval in the binomial poset L*L where L is the binomial poset of all finite dimensional subspaces over a countably infinite dimensional subspace and * is the Segre product (see Stanley reference). T(n,k) is the number of elements at rank k in an n-interval of L*L.

			 1
 1,   1
 1,   9,     1
 1,  49,    49,     1
 1, 225,  1225,   225,   1
 1, 961, 24025, 24025, 961, 1

R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, section 3.18.

Wikipedia, Green's relations

Cf. A243950 (row sums), A022166, A005329, A002884, A002416, A296548 (a subposet of L*L).

nn = 8; B[n_, q_] := QFactorial[n, q]^2; e[x_] := Sum[x^n/B[n, 2], {n, 0, nn}]; Map[Select[#, # > 0 &] &, Table[QFactorial[n, 2]^2, {n, 0, nn}] CoefficientList[
   Series[e[x] e[y x], {x, 0, nn}], {x, y}]]

T(n,k) = A022166(n,k)^2.
Sum_{k=0..n} T(n,k)*A002884(k) = A002416(n).
Let B(n) = A005329(n)^2.  Let E(x)=Sum_{n>=0}x^n/B(n).  Then Sum{n>=0} Sum{k=0..n} T(n,k)*y^k*x^n/B(n) = E(x)*E(y*x).

A372230 Triangular array read by rows. T(n,k) is the number of size k circuits in the linear matroid M[A] where A is the n X 2^n-1 matrix whose columns are the nonzero vectors in GF(2)^n, n>=2, 3<=k<=n+1.

Original entry on oeis.org

1, 7, 7, 35, 105, 168, 155, 1085, 5208, 13888, 651, 9765, 109368, 874944, 3999744, 2667, 82677, 1984248, 37039296, 507967488, 4063739904, 10795, 680085, 33732216, 1349288640, 43177236480, 1036253675520, 14737830051840

Offset: 2

Geoffrey Critzer, Apr 28 2024

For n>=2 and 3<=k<=n, to construct a size k circuit of M[A]: Choose a basis b_1,b_2,...,b_{k-1} of a k-1 dimensional subspace of GF(2)^n. Append the vector b_1 + b_2 + ... + b_{k-1}.

			Triangle begins ...
    1;
    7,    7;
   35,   105,     168;
  155,  1085,    5208,    13888;
  651,  9765,  109368,   874944,   3999744;
 2667, 82677, 1984248, 37039296, 507967488, 4063739904;
...

J. Oxley, Matroid Theory, Oxford Graduate Texts in Mathematics, 1992, page 8.

Cf. A022166, A053601, A006095, A372350 (row sums).

nn = 8; Map[Select[#, # > 0 &] &, Table[Table[PadRight[Table[Product[(2^n - 2^i)/(2^k - 2^i), {i, 0, k - 1}], {k, 2, n}], nn], {n, 2, nn}][[All, j]]*    Table[Product[2^n - 2^i, {i, 0, n - 1}]/(n + 1)!, {n, 2, nn}][[j]], {j, 1, nn - 1}] // Transpose] // Grid

T(n,k) = A022166(n,k-1)*A053601(k-1)/k.
T(n,3) = A006095.
T(n,n+1) = A053601(n)/(n+1).

A385434 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2, reduced mod 3.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1

Offset: 0

David Radcliffe, Jun 28 2025

Row sums give A385435.

			Triangle begins:
  [ 0] [1]
  [ 1] [1, 1]
  [ 2] [1, 0, 1]
  [ 3] [1, 1, 1, 1]
  [ 4] [1, 0, 2, 0, 1]
  [ 5] [1, 1, 2, 2, 1, 1]
  [ 6] [1, 0, 0, 0, 0, 0, 1]
  [ 7] [1, 1, 0, 0, 0, 0, 1, 1]
  [ 8] [1, 0, 1, 0, 0, 0, 1, 0, 1]
  [ 9] [1, 1, 1, 1, 0, 0, 1, 1, 1, 1]
  [10] [1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1]
  [11] [1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1]

Donald E. Knuth and Herbert S. Wilf, The power of a prime that divides a generalized binomial coefficient, J. Reine Angew. Math., 396:212-219, 1989.
Romeo Meštrović, Lucas' theorem: its generalizations, extensions and applications (1878--2014), arXiv preprint arXiv:1409.3820 [math.NT], 2014. (See Equation (80) on p. 31.)

Cf. A007318, A022166, A385435.

Table[Mod[QBinomial[n, k, 2],3], {n, 0, 11}, {k, 0, n}] // Flatten  (* James C. McMahon, Jun 29 2025 *)

def T(n, k): return mod(gaussian_binomial(n, k).subs(q=2), 3)
for n in range(12): print([T(n, k) for k in range(n+1)])  # Peter Luschny, Jun 29 2025

a(n) = A022166(n) mod 3.

T(2n, 2k) = T(2n+1, 2k) = T(2n, 2k+1) = binomial(n, k) mod 3; T(2n, 2k+1) = 0.

A124384 O.g.f.: A(x) = Sum_{n>=0} x^nProduct_{k=0..n} (1 + 2^kx).

Original entry on oeis.org

1, 2, 4, 10, 30, 110, 494, 2734, 18734, 159278, 1685550, 22268974, 367653934, 7597868078, 196929315886, 6402998805550, 261393582040110, 13416320169124910, 865576139256079406, 70227589169019724846, 7172766017169503134766, 921829147482582383174702

Offset: 0

Paul D. Hanna, Oct 28 2006

nonn

			A(x) = (1+x) + x*(1+x)*(1+2x) + x^2*(1+x)*(1+2x)*(1+4x) + x^3*(1+x)*(1+2x)*(1+4x)*(1+8x) +...

Cf. A022166.

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

G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - (1+x*2^k)/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 16 2013

a(n) = Sum_{k=0..floor((n+1)/2)} q-binomial(n-k+1,k)*2^binomial(k,2), where q-binomial is triangle A022166, that is, with q=2. - Vladimir Kruchinin, Jan 21 2020

A276823 a(n) = 3 * [3n]_2! / ([2n+1]_2! * [n+1]_2!), where [n]_q! is the q-factorial.

Original entry on oeis.org

1, 9, 1241, 2634489, 87807053113, 46414431022602681, 390913823614809035461305, 52571422826552549403006580802745, 113007269646365312407427675894837602068665, 3884802624238339577626451297006421856376970743148729

Offset: 1

Vladimir Reshetnikov, Sep 18 2016

nonn

Eric Weisstein's World of Mathematics, q-Factorial, q-Binomial Coefficient.

Cf. A069777, A005329, A022166, A218449.

a:= n-> 3*mul((2^j-1), j=1..3*n)/
         (mul((2^j-1), j=1..2*n+1)*
          mul((2^j-1), j=1..n+1)):
seq(a(n), n=1..12);  # Alois P. Heinz, Sep 20 2016

Table[3 QFactorial[3 n, 2]/(QFactorial[2 n + 1, 2] QFactorial[n + 1, 2]), {n, 10}] (* or *)
Table[3 QBinomial[3 n, 2 n + 1, 2]/(1 - 3 * 2^n + 2^(2 n + 1)), {n, 10}]

a(n) ~ c * 2^((n-2)*(2*n+1)), where c = 3/QPochhammer(1/2, 1/2) = 3*A065446 = 3/A048651. - Vaclav Kotesovec, Sep 20 2016

A277271 Second largest coefficient among the polynomials in row n of the triangle of q-binomial coefficients.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 19, 30, 55, 90, 166, 285, 519, 902, 1656, 2929, 5424, 9673, 18012, 32467, 60981, 110599, 208445, 381301, 722552, 1327869, 2522994, 4665786, 8902311, 16524759, 31594853, 58935171, 113038371, 211499060, 406350261, 763246536, 1470080699

Offset: 4

Vladimir Reshetnikov, Oct 07 2016

nonn

q-binomial coefficients are polynomials in q with integer coefficients.

			Row 5 of the triangle of q-binomial coefficients is [1, 1 + q + q^2 + q^3 + q^4, 1 + q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6, 1 + q + 2*q^2 + 2*q^3 + 2*q^4 + q^5 + q^6, 1 + q + q^2 + q^3 + q^4, 1]. The largest coefficient is 2, and the second largest coefficient is 1. Hence A277218(5) = 2 and a(5) = 1.

Eric W. Weisstein, q-Binomial Coefficient
Wikipedia, q-binomial

Cf. A002838, A022166, A029895, A055606, A076822, A277218 (largest coefficients).

Table[(Union @@ Table[CoefficientList[FunctionExpand[QBinomial[n, k, q]], q], {k, 0, n}])[[-2]], {n, 4, 40}]

A289383 Total number of nonzero vectors over all subspaces of an n-dimensional vector space over the field with two elements.

Original entry on oeis.org

0, 1, 6, 35, 240, 2077, 23562, 358775, 7449060, 213188689, 8473977534, 470309723435, 36582636406680, 3998655357260293, 615328930033081458, 133485330929459963615, 40859530900982506959180, 17659495180812130332490681, 10781678259164073608877557286, 9301770545157096607562560360595

Offset: 0

Geoffrey Critzer, Jul 04 2017

nonn

The q-analog of A001787.

Cf. A001787, A006116, A022166.

nn = 20; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}];Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0, nn}] CoefficientList[Series[z eq[z]^2 /. q -> 2, {z, 0, nn}], z]

a(n) = Sum_{k=1..n} A022166(n,k)*(2^k - 1).
a(n)/[n]_q! is the coefficient of x^n in the expansion of x*exp_q(x)^2 when q->2 and where exp_q(x) is the q exponential function and [n]_q! is the q-factorial of n.
a(n) = (2^n - 1)*A006116(n-1).

A329154 Coefficients of polynomials related to the sum of Gaussian binomial coefficients for q = 2. Triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 26, 24, 12, 4, 1, 158, 130, 60, 20, 5, 1, 1330, 948, 390, 120, 30, 6, 1, 15414, 9310, 3318, 910, 210, 42, 7, 1, 245578, 123312, 37240, 8848, 1820, 336, 56, 8, 1, 5382862, 2210202, 554904, 111720, 19908, 3276, 504, 72, 9, 1

Offset: 0

Vladimir Kruchinin and Peter Luschny, Mar 08 2020

T(n,k) is the number of n X n matrices over F_2 in reduced row echelon form having exactly k zero-columns. Equivalently, T(n,k) is the number of subspaces of F_2^n that "involve" n-k coordinates. (For the definition of "involve" see the link below: D. E. Knuth, Letter to Daniel Ullman and others). - Geoffrey Critzer, May 03 2025

			Triangle starts:
[0] [1]
[1] [1,       1]
[2] [2,       2,       1]
[3] [6,       6,       3,      1]
[4] [26,      24,      12,     4,      1]
[5] [158,     130,     60,     20,     5,     1]
[6] [1330,    948,     390,    120,    30,    6,    1]
[7] [15414,   9310,    3318,   910,    210,   42,   7,   1]
[8] [245578,  123312,  37240,  8848,   1820,  336,  56,  8,  1]
[9] [5382862, 2210202, 554904, 111720, 19908, 3276, 504, 72, 9, 1]

D. E. Knuth, Letter to Daniel Ullman and others, Apr 29 1997 [Annotated scanned copy, with permission]
R. P. Stanley and S. C. Locke, Graphs without increasing paths: Solution to Problem 10572, The American Mathematical Monthly, 106(2) (1999), 168.

Row sums: A006116, first column: A135922.

Cf. A022166, A323842, A323843.

T := (n, k) -> local j, m; pochhammer(n - k + 1, k)*add((-1)^j*add(product((2^(i + m) - 1)/(2^i - 1), i = 1..n-k-m-j), m = 0..n-k-j)*binomial(n - k, j), j = 0..n-k) / k!: for n from 0 to 9 do seq(T(n,k), k=0..n) od;  # Peter Luschny, Oct 08 2023

T[n_,k_]:= (Pochhammer[n-k+1,k]/(k!)*Sum[(-1)^j*Sum[Product[(2^(i+m)-1)/(2^i-1),{i,1,n-k-m-j}],{m,0,n-k-j}]*Binomial[n-k,j],{j,0,n-k}]); Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Oct 07 2023 *)

R = PolynomialRing(QQ, 'x')
x = R.gen()
@cached_function
def P(n, k, x):
    if k < 0 or n < 0: return R(0)
    if k == 0: return R(1)
    return x*P(n, k-1, x) + 2^k*P(n-1, k, (x+1)/2)
def row(n): return sum(P(n-k, k, x) for k in range(n+1)).coefficients()
print(flatten([row(n) for n in range(10)]))

Let P(n, k, x) = x*P(n, k-1, x) + 2^k*P(n-1, k, (x+1)/2) and Q(n, x) = Sum_{k=0..n} P(n-k, k, x) then T(n, k) = [x^k] Q(n, x).
T(n, k) = (1/k!) * Pochhammer(n-k+1, k) * Sum_{j=0..n-k}((-1)^j*Sum_{m=0..n-k-j} (Product_{i=1..n-k-m-j} ((2^(i+m)-1)/(2^i-1))) * binomial(n-k, j)). - Detlef Meya, Oct 07 2023
T(n,k) = binomial(n,k)*A135922(n-k). (see Stanley-Locke link above) - Geoffrey Critzer, May 03 2025
E.g.f.: exp(y x)*f(x) where f(x) is the e.g.f. for A135922. - Geoffrey Critzer, May 03 2025

A355401 Triangle read by rows: T(n, k) = Sum_{i=1..n-k} inverse-q-binomial(n-k-1, i-1) * q-binomial(n-2+i, n-2) for 0 < k < n with initial values T(n, 0) = 0 for n > 0 and T(n, n) = 1 for n >= 0, here q = 2.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 3, 1, 0, 64, 28, 7, 1, 0, 4096, 960, 140, 15, 1, 0, 1048576, 126976, 9920, 620, 31, 1, 0, 1073741824, 66060288, 2666496, 89280, 2604, 63, 1, 0, 4398046511104, 136365211648, 2796552192, 48377856, 755904, 10668, 127, 1

Offset: 0

Werner Schulte, Jun 30 2022

The Gaussian or q-binomial coefficients [n, k]_q for 0 <= k <= n are the basis for lower triangular matrices T_q, which are created by an unusual formula. This triangle is the result for q = 2. The general construction is as follows:
  For some fixed integer q define the infinite lower triangular matrix M_q by M(q; n, 0) = 0 for n > 0, and M(q; n, n) = 1 for n >= 0, and M(q; n, k) = M(q; n-1, k-1) + q^(k-1) * M(q; n-1, k) for 0 < k < n. Then the matrix inverse I_q = M_q^(-1) exists, and M(q; n, k) = [n-1, k-1]q for 0 < k <= n. Next define the triangle T_q by 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 and T(q; n, 0) = 0^n for n >= 0. For q = 1 see A097805 and for q = 2 see this triangle.
Conjecture: T(q; n+1, 1) = q^(n*n-n) for n >= 0.
Conjecture: T(q; n, k) = q^((n-k-1)*(n-k)) * M(q; n, k) for 0 <= k <= n.
Conjecture: Define g(q; n) = -Sum_{i=0..n-1} [n, i]_q * g(q; i) * T(q; n+1-i, 1) for n > 0 with g(q; 0) = 1. Then the matrix inverse R_q = T_q^(-1) is given by R(q; n, k) = g(q; n-k) * M(q; n, k) for 0 <= k <= n, and g(q; n) = R(q; n+1, 1) for n >= 0.

			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              4             3           1
  4 :  0             64            28           7         1
  5 :  0           4096           960         140        15       1
  6 :  0        1048576        126976        9920       620      31      1
  7 :  0     1073741824      66060288     2666496     89280    2604     63    1
  8 :  0  4398046511104  136365211648  2796552192  48377856  755904  10668  127  1
  etc.
Matrix inverse R(n, k) for 0 <= k <= n starts:
n\k :  0           1          2         3       4     5    6  7
===============================================================
  0 :  1
  1 :  0           1
  2 :  0          -1          1
  3 :  0          -1         -3         1
  4 :  0         -29         -7        -7       1
  5 :  0       -2561       -435       -35     -15     1
  6 :  0     -814309     -79391     -4495    -155   -31    1
  7 :  0  -944455609  -51301467  -1667211  -40455  -651  -63  1
  etc.

Cf. A022166, A053763 (column 1), A135950.

Conjecture: T(n+1, 1) = A053763(n) = 2^(n*n - n) for n >= 0.
Conjecture: T(n, k) = 2^((n-k-1) * (n-k)) * A022166(n-1, k-1) for 0 < k <= n, and T(n, 0) = 0^n for n >= 0.
Conjecture: Define g(n) = -Sum_{i=0..n-1} A022166(n, i) * g(i) * T(n+1-i, 1) for n > 0 with g(0) = 1. Then matrix inverse R = T^(-1) is given by R(n, 0) = 0^n for n >= 0, and R(n, k) = g(n-k) * A022166(n-1, k-1) for 0 < k <= n, and g(n) = R(n+1, 1) for n >= 0.

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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A359313 Triangular array read by rows. T(n,k) is the number of Green's H-classes contained in the D-class of rank k matrices in the semigroup Mat_n(F_2) of n X n matrices over the field F_2. n>=0, 0<=k<=n.

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A372230 Triangular array read by rows. T(n,k) is the number of size k circuits in the linear matroid M[A] where A is the n X 2^n-1 matrix whose columns are the nonzero vectors in GF(2)^n, n>=2, 3<=k<=n+1.

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A385434 Triangle of Gaussian binomial coefficients (or q-binomial coefficients) [n,k] for q = 2, reduced mod 3.

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A124384 O.g.f.: A(x) = Sum_{n>=0} x^n*Product_{k=0..n} (1 + 2^k*x).

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A276823 a(n) = 3 * [3*n]_2! / ([2*n+1]_2! * [n+1]_2!), where [n]_q! is the q-factorial.

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A277271 Second largest coefficient among the polynomials in row n of the triangle of q-binomial coefficients.

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A289383 Total number of nonzero vectors over all subspaces of an n-dimensional vector space over the field with two elements.

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A124384 O.g.f.: A(x) = Sum_{n>=0} x^nProduct_{k=0..n} (1 + 2^kx).

A276823 a(n) = 3 * [3n]_2! / ([2n+1]_2! * [n+1]_2!), where [n]_q! is the q-factorial.