A022168 -id:A022168 - cp's OEIS frontend

A347487 Irregular triangle read by rows: T(n, k) is the q-multinomial coefficient defined by the k-th partition of n in Abramowitz-Stegun order, evaluated at q = 4.

1, 1, 5, 1, 21, 105, 1, 85, 357, 1785, 8925, 1, 341, 5797, 28985, 121737, 608685, 3043425, 1, 1365, 93093, 376805, 465465, 7912905, 33234201, 39564525, 166171005, 830855025, 4154275125, 1, 5461, 1490853, 24208613, 7454265, 508380873, 2057732105, 8642474841

Offset: 1

Álvar Ibeas, Sep 03 2021

Abuse of notation: we write T(n, L) for T(n, k), where L is the k-th partition of n in A-St order.

For any permutation (e_1,...,e_r) of the parts of L, T(n, L) is the number of chains of subspaces 0 < V_1 < ··· < V_r = (F_4)^n with dimension increments (e_1,...,e_r).

			The number of subspace chains 0 < V_1 < V_2 < (F_4)^3 is 105 = T(3, (1, 1, 1)). There are 21 = A022168(3, 1) choices for a one-dimensional subspace V_1 and, for each of them, 5 = A022168(2, 1) extensions to a two-dimensional subspace V_2.
Triangle begins:
  k:  1   2    3     4      5      6       7
      --------------------------------------
n=1:  1
n=2:  1   5
n=3:  1  21  105
n=4:  1  85  357  1785   8925
n=5:  1 341 5795 28985 121737 608685 3043425

R. P. Stanley, Enumerative Combinatorics (vol. 1), Cambridge University Press (1997), Section 1.3.

Álvar Ibeas, First 20 rows, flattened

Cf. A036038 (q = 1), A022168, A015002 (last entry in each row).

T(n, (n)) = 1. T(n, L) = A022168(n, e) * T(n - e, L \ {e}), if L is a partition of n and e < n is a part of L.

A347971 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_4)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 12, 31, 12, 1, 1, 19, 111, 111, 19, 1, 1, 29, 361, 964, 361, 29, 1, 1, 41, 1068, 8042, 8042, 1068, 41, 1, 1, 56, 2954, 64674, 205065, 64674, 2954, 56, 1, 1, 75, 7681, 492387, 5402621, 5402621, 492387, 7681, 75, 1, 1, 97, 18880, 3507681, 137287827

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.

Regarding the formula for column k = 1, note that A241926(q - 1, n) counts, up to coordinate permutation, one-dimensional subspaces of (F_q)^n generated by a vector with no zero component.

			Triangle begins:
  k:  0    1    2    3    4    5    6
      -------------------------------
n=0:  1
n=1:  1    1
n=2:  1    3    1
n=3:  1    7    7    1
n=4:  1   12   31   12    1
n=5:  1   19  111  111   19    1
n=6:  1   29  361  964  361   29    1
There are 5 = A022168(2, 1) one-dimensional subspaces in (F_4)^2, namely, those generated by vectors (0, 1), (1, 0), (1, 1), (1, x), and (1, x + 1), where F_4 = F_2[x] / (x^2 + x + 1). The coordinate swap identifies the first two on the one hand and the last two on the other, while <(1, 1)> is invariant. Hence, T(2, 1) = 3.

Álvar Ibeas, Entries up to T(14, 6)
H. Fripertinger, Isometry classes of codes
H. Fripertinger, Number of the isometry classes of all quaternary (n,k)-codes
Álvar Ibeas, Column k=1 up to n=100
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

Cf. A022168, A007997, A241926.

T(n, 1) = T(n - 1, 1) + A007997(n + 5).

A383755 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = 3^(n-k) * T(n-1,k-1) + 4^k * T(n-1,k) with T(n,k) = n^k if n*k=0.

Original entry on oeis.org

1, 1, 1, 1, 7, 1, 1, 37, 37, 1, 1, 175, 925, 175, 1, 1, 781, 19525, 19525, 781, 1, 1, 3367, 375661, 1776775, 375661, 3367, 1, 1, 14197, 6828757, 144142141, 144142141, 6828757, 14197, 1, 1, 58975, 119609725, 10884484975, 48575901517, 10884484975, 119609725, 58975, 1

Offset: 0

Seiichi Manyama, May 09 2025

			Triangle begins:
  1;
  1,    1;
  1,    7,      1;
  1,   37,     37,       1;
  1,  175,    925,     175,      1;
  1,  781,  19525,   19525,    781,    1;
  1, 3367, 375661, 1776775, 375661, 3367, 1;
  ...

Columns k=0..3 give A000012, A005061, A383756(n-2), A383757(n-3).

Cf. A022168.

T(n, k) = if(n*k==0, n^k, 3^(n-k)*T(n-1, k-1)+4^k*T(n-1, k));

def a_row(n): return [3^(k*(n-k))*q_binomial(n, k, 4/3) for k in (0..n)]
for n in (0..8): print(a_row(n))

T(n,k) = 3^(k*(n-k)) * q-binomial(n, k, 4/3).
T(n,k) = 4^(n-k) * T(n-1,k-1) + 3^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) = 4^j - 3^j.

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

A173583 Triangle T(n, k, q) = q-binomial(n, k, q^2), for q = 5, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 26, 1, 1, 651, 651, 1, 1, 16276, 407526, 16276, 1, 1, 406901, 254720026, 254720026, 406901, 1, 1, 10172526, 159200423151, 3980255126276, 159200423151, 10172526, 1, 1, 254313151, 99500274641901, 62191645548485651, 62191645548485651, 99500274641901, 254313151, 1

Offset: 0

Roger L. Bagula, Feb 22 2010

Row sums are: 1, 2, 28, 1304, 440080, 510253856, 4298676317632, 124582292154881408, ...

			Triangle begins as:
  1;
  1,        1;
  1,       26,            1;
  1,      651,          651,             1;
  1,    16276,       407526,         16276,            1;
  1,   406901,    254720026,     254720026,       406901,        1;
  1, 10172526, 159200423151, 3980255126276, 159200423151, 10172526, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000012 (q=0), A007318 (q=1), A022168 (q=2), A022173 (q=3), A022180 (q=4), A173583 (q=5).

q:=5;; [q^(k*(n-k))*GaussianBinomial(n, k, q): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 22 2021

(* First program *)
c[n_, q_]:= Product[(1 -q^(2*j))/(1-q), {j,1,n}];
T[n_, k_, q_]:= c[n, q]/(c[k, q]*c[n-k, q]);
Table[T[n, k, 5], {n,0,12}, {k,0,n}]//Flatten
(* Second program *)
Table[QBinomial[n,k,5^2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 22 2021 *)
T[n_, k_, p_]:= T[n, k, p] = If[k==0 || k==n, 1, T[n-1, k-1, p] + p^k*T[n-1, k, q]];  Table[T[n, k, 25], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Feb 22 2021 *)

flatten([[q_binomial(n, k, 5^2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 22 2021

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)) where c(n, q) = Product_{j=1..n} (1 -q^(2*j))/(1-q) for q = 5.
From G. C. Greubel, Feb 22 2021: (Start)
T(n, k, q) = q-binomial(n, k, q^2), for q = 5.
T(n, k) = T(n-1, k-1) + p^k * T(n-1, k), with p = 25 (as a number triangle). (End)

Edited by G. C. Greubel, Feb 22 2021

A211877 Number of involutions in GL(n,4).

Original entry on oeis.org

1, 21, 673, 102273, 47663617, 110981851137, 815432848809985, 30052835284679819265, 3519512226295269640765441, 2069751512310185039905834926081, 3874510079394593253089862950754189313, 36431456010689490638771956423547489198538753

Offset: 1

Alexander Gruber, Feb 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..50

A211877 := proc(n)
    add( 4^(k*(n-k))*A022168(n,k),k=1..n) ;
end proc: # R. J. Mathar, Apr 26 2013

Sum[q^(k (n - k)) QBinomial[n, k, q], {k, 1, n}]

cp's OEIS Frontend

A347487 Irregular triangle read by rows: T(n, k) is the q-multinomial coefficient defined by the k-th partition of n in Abramowitz-Stegun order, evaluated at q = 4.

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A347971 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_4)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

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A383755 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) = 3^(n-k) * T(n-1,k-1) + 4^k * T(n-1,k) with T(n,k) = n^k if n*k=0.

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

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Magma

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A173583 Triangle T(n, k, q) = q-binomial(n, k, q^2), for q = 5, read by rows.

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Magma

Mathematica

Sage

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Extensions

A211877 Number of involutions in GL(n,4).

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Maple

Mathematica