A022167 -id:A022167 - cp's OEIS frontend

A347970 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_3)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 8, 16, 8, 1, 1, 11, 39, 39, 11, 1, 1, 15, 87, 168, 87, 15, 1, 1, 19, 176, 644, 644, 176, 19, 1, 1, 24, 338, 2348, 4849, 2348, 338, 24, 1, 1, 29, 613, 8137, 37159, 37159, 8137, 613, 29, 1, 1, 35, 1071, 27047, 286747, 679054, 286747, 27047, 1071

Offset: 0

Álvar Ibeas, Sep 21 2021

Columns can be computed by a method analogous to that of Fripertinger for isometry classes of linear codes, disallowing scalar transformation of individual coordinates.

			Triangle begins:
  k:  0   1   2   3   4   5   6   7
      -----------------------------
n=0:  1
n=1:  1   1
n=2:  1   3   1
n=3:  1   5   5   1
n=4:  1   8  16   8   1
n=5:  1  11  39  39  11   1
n=6:  1  15  87 168  87  15   1
n=7:  1  19 176 644 644 176  19   1
There are 4 = A022167(2, 1) one-dimensional subspaces in (F_3)^2, namely, those generated by (0, 1), (1, 0), (1, 1), and (1, 2). The first two are related by coordinate swap, while the remaining two are invariant. Hence, T(2, 1) = 3.

Álvar Ibeas, Entries up to T(16, 7)
H. Fripertinger, Isometry classes of codes
H. Fripertinger, Number of the isometry classes of all ternary (n,k)-codes
Álvar Ibeas, Column k=2 up to n=100
Álvar Ibeas, Column k=3 up to n=100
Álvar Ibeas, Column k=4 up to n=100
Álvar Ibeas, Column k=5 up to n=100
Álvar Ibeas, Column k=6 up to n=100
Álvar Ibeas, Column k=7 up to n=100

Cf. A022167, A024206(n+1) (column k=1), A076831.

A378666 Triangular array read by rows: T(n,k) is the number of n X n idempotent matrices over GF(3) having rank k, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 117, 117, 1, 1, 1080, 10530, 1080, 1, 1, 9801, 882090, 882090, 9801, 1, 1, 88452, 72243171, 666860040, 72243171, 88452, 1, 1, 796797, 5873190687, 491992666011, 491992666011, 5873190687, 796797, 1, 1, 7173360, 476309310660, 360089838858960, 3267815287645062, 360089838858960, 476309310660, 7173360, 1

Offset: 0

Geoffrey Critzer, Dec 02 2024

A matrix M is idempotent if M^2 = M.

			Triangle T(n,k) begins:
  1;
  1,    1;
  1,   12,      1;
  1,  117,    117,      1;
  1, 1080,  10530,   1080,    1;
  1, 9801, 882090, 882090, 9801, 1;
  ...

Cf. A296548, A053846 (row sums).

Cf. A022167, A118180.

b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, b(n-1, k-1)+3^k*b(n-1, k)))
    end:
T:= (n,k)-> 3^(k*(n-k))*b(n, k):
seq(seq(T(n,k), k=0..n), n=0..8);  # Alois P. Heinz, Dec 02 2024

nn = 8; \[Gamma][n_, q_] := Product[q^n - q^i, {i, 0, n - 1}]; B[n_, q_] := \[Gamma][n, q]/(q - 1)^n; \[Zeta][x_] := Sum[x^n/B[n, 3], {n, 0, nn}];Map[Select[#, # > 0 &] &, Table[B[n, 3], {n,0,nn}]*CoefficientList[Series[\[Zeta][x] \[Zeta][y x], {x, 0, nn}], {x, y}]] // Flatten

Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/B(n) = e(x)*e(y*x) where e(x) = Sum_{n>=0} x^n/B(n) and B(n) = A053290(n)/2^n.

T(n,k) = A022167(n,k) * A118180(n,k). - Alois P. Heinz, Dec 02 2024

A006103 Gaussian binomial coefficient [ 2n,n ] for q=3.

Original entry on oeis.org

1, 4, 130, 33880, 75913222, 1506472167928, 267598665689058580, 427028776969176679964080, 6129263888495201102915629695046, 791614563787525746761491781638123230424, 920094266641283414155073889843358388073398779900

Offset: 0

N. J. A. Sloane

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.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

T. D. Noe, Table of n, a(n) for n = 0..25
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)

Cf. A022167.

q:=3; [n le 0 select 1 else (&*[(1-q^(2*n-j))/(1-q^(j+1)): j in [0..n-1]]): n in [0..15]]; // G. C. Greubel, Mar 09 2019

Table[QBinomial[2n, n, 3], {n, 0, 10}] (* Vladimir Reshetnikov, Sep 12 2016 *)

q=3; {a(n) = prod(j=0, n-1, (1-q^(2*n-j))/(1-q^(j+1))) };
vector(15, n, n--; a(n)) \\ G. C. Greubel, Mar 09 2019

[gaussian_binomial(2*n,n,3) for n in (0..15)] # G. C. Greubel, Mar 09 2019

a(n) = Sum_{k=0..n} 3^(k^2)*(A022167(n,k))^2. - Werner Schulte, Mar 09 2019

A156914 Square array T(n, k) = q-binomial(2*n, n, k+1), read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 35, 20, 1, 5, 130, 1395, 70, 1, 6, 357, 33880, 200787, 252, 1, 7, 806, 376805, 75913222, 109221651, 924, 1, 8, 1591, 2558556, 6221613541, 1506472167928, 230674393235, 3432, 1, 9, 2850, 12485095, 200525284806, 1634141006295525, 267598665689058580, 1919209135381395, 12870

Offset: 0

Roger L. Bagula, Feb 18 2009

			Square array begins as:
    1,         1,             1,                1, ...;
    2,         3,             4,                5, ...;
    6,        35,           130,              357, ...;
   20,      1395,         33880,           376805, ...;
   70,    200787,      75913222,       6221613541, ...;
  252, 109221651, 1506472167928, 1634141006295525, ...;
Antidiagonal triangle begins as:
  1;
  1, 2;
  1, 3,    6;
  1, 4,   35,      20;
  1, 5,  130,    1395,         70;
  1, 6,  357,   33880,     200787,           252;
  1, 7,  806,  376805,   75913222,     109221651,          924;
  1, 8, 1591, 2558556, 6221613541, 1506472167928, 230674393235, 3432;

G. C. Greubel, Antidiagonal rows n = 0..25, flattened

Cf. A000984, A022166, A022167, A022168, A022169, A022170, A022171, A022175.

QBinomial:= func< n,k,q | q eq 1 select Binomial(n, k) else k eq 0 select 1 else (&*[ (1-q^(n-j+1))/(1-q^j): j in [1..k] ]) >;
T:= func< n,k | QBinomial(2*n, n, k+1) >;
[T(k, n-k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 14 2021

T[n_, k_]:= QBinomial[2*n, n, k+1];
Table[T[k, n-k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 14 2021 *)

def A156914(n, k): return q_binomial(2*n, n, k+1)
flatten([[A156914(k,n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 14 2021

T(n, k) = q-binomial(2*n, n, k+1), where q-binomial(n, k, q) = Product_{j=0..k-1} ( (1-q^(n-j))/(1-q^(j+1)) ), read by antidiagonals. - G. C. Greubel, Jun 14 2021

Edited by G. C. Greubel, Jun 14 2021

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.

A383753 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = 2^(n-k) * T(n-1,k-1) + 3^k * T(n-1,k) with T(n,k) = n^k if n*k=0.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 19, 19, 1, 1, 65, 247, 65, 1, 1, 211, 2743, 2743, 211, 1, 1, 665, 28063, 96005, 28063, 665, 1, 1, 2059, 273847, 3041143, 3041143, 273847, 2059, 1, 1, 6305, 2596399, 90873965, 294990871, 90873965, 2596399, 6305, 1, 1, 19171, 24174631, 2619766591, 26802227431, 26802227431, 2619766591, 24174631, 19171, 1

Offset: 0

Seiichi Manyama, May 09 2025

			Triangle begins:
  1;
  1,    1;
  1,    5,      1;
  1,   19,     19,       1;
  1,   65,    247,      65,       1;
  1,  211,   2743,    2743,     211,      1;
  1,  665,  28063,   96005,   28063,    665,    1;
  1, 2059, 273847, 3041143, 3041143, 273847, 2059, 1;
  ...

Columns k=0..3 give A000012, A001047, A019443(n-2), A383754(n-3).

Cf. A022167.

T(n, k) = if(n*k==0, n^k, 2^(n-k)*T(n-1, k-1)+3^k*T(n-1, k));

def a_row(n): return [2^(k*(n-k))*q_binomial(n, k, 3/2) for k in (0..n)]
for n in (0..9): print(a_row(n))

T(n,k) = 2^(k*(n-k)) * q-binomial(n, k, 3/2).
T(n,k) = 3^(n-k) * T(n-1,k-1) + 2^k * T(n-1,k).
T(n,k) = T(n,n-k).
G.f. of column k: x^k * exp( Sum_{j>=1} f((k+1)*j)/f(j) * x^j/j ), where f(j) = 3^j - 2^j.

cp's OEIS Frontend

A347970 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_3)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

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A378666 Triangular array read by rows: T(n,k) is the number of n X n idempotent matrices over GF(3) having rank k, n>=0, 0<=k<=n.

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A006103 Gaussian binomial coefficient [ 2n,n ] for q=3.

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A156914 Square array T(n, k) = q-binomial(2*n, n, k+1), read by antidiagonals.

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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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A383753 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = 2^(n-k) * T(n-1,k-1) + 3^k * T(n-1,k) with T(n,k) = n^k if n*k=0.

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