A047927 -id:A047927 - cp's OEIS frontend

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

1, 2, 1, 3, 4, 2, 4, 9, 12, 6, 5, 16, 36, 48, 24, 6, 25, 80, 180, 240, 120, 7, 36, 150, 480, 1080, 1440, 720, 8, 49, 252, 1050, 3360, 7560, 10080, 5040, 9, 64, 392, 2016, 8400, 26880, 60480, 80640, 40320, 10, 81, 576, 3528, 18144, 75600, 241920, 544320

Offset: 0

Reinhard Zumkeller, Aug 30 2014

row(0) = {1}; row(n+1) = row(n) multiplied by n and prepended with (n+1);
A111063(n+1) = sum of n-th row;
T(2*n,n) = A002690(n), central terms;
T(n,0) = n + 1;
T(n,1) = A000290(n), n > 0;
T(n,2) = A011379(n-1), n > 1;
T(n,3) = A047927(n), n > 2;
T(n,4) = A192849(n-1), n > 3;
T(n,5) = A000142(5) * A027810(n-5), n > 4;
T(n,6) = A000142(6) * A027818(n-6), n > 5;
T(n,7) = A000142(7) * A056001(n-7), n > 6;
T(n,8) = A000142(8) * A056003(n-8), n > 7;
T(n,9) = A000142(9) * A056114(n-9), n > 8;
T(n,n-10) = 11 * A051431(n-10), n > 9;
T(n,n-9) = 10 * A049398(n-9), n > 8;
T(n,n-8) = 9 * A049389(n-8), n > 7;
T(n,n-7) = 8 * A049388(n-7), n > 6;
T(n,n-6) = 7 * A001730(n), n > 5;
T(n,n-5) = 6 * A001725(n), n > 5;
T(n,n-4) = 5 * A001720(n), n > 4;
T(n,n-3) = 4 * A001715(n), n > 2;
T(n,n-2) = A070960(n), n > 1;
T(n,n-1) = A052849(n), n > 0;
T(n,n) = A000142(n);
T(n,k) = A137948(n,k) * A007318(n,k), 0 <= k <= n.

			.  0:   1;
.  1:   2,  1;
.  2:   3,  4,   2;
.  3:   4,  9,  12,    6;
.  4:   5, 16,  36,   48,    24;
.  5:   6, 25,  80,  180,   240,   120;
.  6:   7, 36, 150,  480,  1080,  1440,    720;
.  7:   8, 49, 252, 1050,  3360,  7560,  10080,   5040;
.  8:   9, 64, 392, 2016,  8400, 26880,  60480,  80640,  40320;
.  9:  10, 81, 576, 3528, 18144, 75600, 241920, 544320, 725760, 362880.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Cf. A111063 (row sums), A240993 (row products), A002690 (central terms).
Cf. A000290, A011379, A027810, A027818, A047927, A056001, A056003, A056114, A192849.
Cf. A000142, A001715, A001720, A001725, A001730, A049388, A049389, A049398, A051431, A052849, A070960.
Cf. A007318, A137948.

a245334 n k = a245334_tabl !! n !! k
a245334_row n = a245334_tabl !! n
a245334_tabl = iterate (\row@(h:_) -> (h + 1) : map (* h) row) [1]

Table[(n!)/((n - k)!)*(n + 1 - k), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Sep 10 2017 *)

T(n,k) = n!*(n+1-k)/(n-k)!. - Werner Schulte, Sep 09 2017

A212163 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the rhombic hexagonal square grid graph RH_(k,k).

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 48, 5, 0, 0, 6, 1056, 180, 6, 0, 0, 6, 45696, 32940, 480, 7, 0, 0, 6, 4034304, 30847500, 393600, 1050, 8, 0, 0, 6, 739642368, 148039757460, 3312560640, 2735250, 2016, 9

Offset: 1

Alois P. Heinz, May 02 2012

The rhombic hexagonal square grid graph RH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges; see A212162 for example. The chromatic polynomial of RH_(n,n) has n^2+1 = A002522(n) coefficients.

A differs from A212195 first at (n,k) = (4,5): A(4,5) = 4034304, A212195(4,5) = 4038432.

			Square array A(n,k) begins:
  1,    0,       0,            0,                 0, ...
  2,    0,       0,            0,                 0, ...
  3,    6,       6,            6,                 6, ...
  4,   48,    1056,        45696,           4034304, ...
  5,  180,   32940,     30847500,      148039757460, ...
  6,  480,  393600,   3312560640,   286169360240640, ...
  7, 1050, 2735250, 123791435250, 97337270132408250, ...

Andrew Howroyd, Table of n, a(n) for n = 1..153
Wikipedia, Chromatic Polynomial

Columns k=1-6 give: A000027, A047927(n) = 6*A002417(n-2), 6*A068244, 6*A068245, 6*A068246, 6*A068247.
Rows n=1-15 give: A000007, A000038, A040006, 4*A068271, 5*A068272, 6*A068273, 7*A068274, 8*A068275, 9*A068276, 10*A068277, 11*A068278, 12*A068279, 13*A068280, 14*A068281, 15*A068282.
Cf. A212162, A212195, A208054, A208050.

A212195 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the staggered hexagonal square grid graph SH_(k,k).

Original entry on oeis.org

1, 0, 2, 0, 0, 3, 0, 0, 6, 4, 0, 0, 6, 48, 5, 0, 0, 6, 1056, 180, 6, 0, 0, 6, 45696, 32940, 480, 7, 0, 0, 6, 4038432, 30847500, 393600, 1050, 8, 0, 0, 6, 743601024, 148046704020, 3312560640, 2735250, 2016, 9

Offset: 1

Alois P. Heinz, May 03 2012

The staggered hexagonal square grid graph SH_(n,n) has n^2 = A000290(n) vertices and (n-1)*(3*n-1) = A045944(n-1) edges; see A212194 for example. The chromatic polynomial of SH_(n,n) has n^2+1 = A002522(n) coefficients.

A differs from A212163 first at (n,k) = (4,5): A(4,5) = 4038432, A212163(4,5) = 4034304.

			Square array A(n,k) begins:
  1,    0,       0,            0,                 0, ...
  2,    0,       0,            0,                 0, ...
  3,    6,       6,            6,                 6, ...
  4,   48,    1056,        45696,           4038432, ...
  5,  180,   32940,     30847500,      148046704020, ...
  6,  480,  393600,   3312560640,   286170443437440, ...
  7, 1050, 2735250, 123791435250, 97337320223288250, ...

Wikipedia, Chromatic Polynomial

Columns k=1-6 give: A000027, A047927(n) = 6*A002417(n-2), 6*A068244, 6*A068245, 6*A068248, 6*A068249.
Rows n=1-10, 16-18 give: A000007, A000038, A040006, 4*A068283, 5*A068284, 6*A068285, 7*A068286, 8*A068287, 9*A068288, 10*A068289, 16*A068290, 17*A068291, 18*A068292.
Cf. A212163, A212194.

A059238 Orders of the finite groups GL_2(K) when K is a finite field with q = A246655(n) elements.

Original entry on oeis.org

6, 48, 180, 480, 2016, 3528, 5760, 13200, 26208, 61200, 78336, 123120, 267168, 374400, 511056, 682080, 892800, 1014816, 1822176, 2755200, 3337488, 4773696, 5644800, 7738848, 11908560, 13615200, 16511040, 19845936, 25048800, 28003968

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Jan 21 2001

nonn

From Jianing Song, Nov 06 2019: (Start)
GL_2(K) means the group of invertible 2 X 2 matrices A over K.
In general, let R be any commutative ring with unity, GL_n(R) be the group of n X n matrices A over R such that det(A) != 0 and SL_n(R) be the group of n X n matrices A over R such that det(A) = 1, then GL_n(R)/SL_n(R) = R* is the multiplicative group of R. This is because if we define f(M) = det(M) for M in GL_n(R), then f is a surjective homomorphism from GL_n(K) to R*, and SL_n(R) is its kernel. Thus |GL_n(R)|/|SL_n(R)| = |R*|; if K is a finite field, then |GL_n(R)|/|SL_n(R)| = |K|-1. (End)

			a(4) = 480 because A246655(4) = 5, and (5^2-1)*(5^2-5) = 480.

Jianing Song, Table of n, a(n) for n = 1..10000
R. A. Wilson, The classical groups, chapter 3.3.1 in The finite Simple Groups, Graduate Texts in Mathematics 251 (2009).

Subsequence of A047927.
Cf. A246655, A000252 (order of GL_2(Z_n)).
For the order of SL_2(K) see A329119.

with(numtheory): for n from 2 to 400 do if nops(ifactors(n)[2]) = 1 then printf(`%d,`, (n+1)*(n)*(n-1)^2) fi: od:

nn=30;a=Take[Union[Sort[Flatten[Table[Table[Prime[m]^k,{m,1,nn}],{k,1,nn}]]]],nn];Table[(q^2-1)(q^2-q),{q,a}]  (* Geoffrey Critzer, Apr 20 2013 *)

[(p+1)*p*(p-1)^2 | p <- [1..200], isprimepower(p)] \\ Jianing Song, Nov 05 2019

If the finite field K has p^m elements, then the order of the group GL_2(K) is (p^(2m)-1)*(p^(2m)-p^m) = (p^m+1)*(p^m)*(p^m-1)^2.
a(n) = A047927(A246655(n)+1). - Jianing Song, Nov 05 2019
a(n) = (A246655(n)-1)*A329119(n). - Jianing Song, Nov 06 2019

More terms from James Sellers, Jan 22 2001

Offset corrected by Jianing Song, Nov 05 2019

A244509 Order of GL_2(p), the general linear group over F_p, where p runs through the primes.

Original entry on oeis.org

6, 48, 480, 2016, 13200, 26208, 78336, 123120, 267168, 682080, 892800, 1822176, 2755200, 3337488, 4773696, 7738848, 11908560, 13615200, 19845936, 25048800, 28003968, 38450880, 46879728, 62029440, 87607296, 103020000, 111447648, 129843216, 139851360

Offset: 1

John McGee, Nov 15 2014

nonn

			For n=3 (p=5) we have a(3) = 4*5*(25-1) = 480.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Abbey Bourdon, Ozlem Ejder, Yuan Liu, Frances Odumodu, Bianca Viray, On the level of modular curves that give rise to sporadic j-invariants, arXiv:1808.04520 [math.NT], 2018. See Table 7.2 (an extract of current sequence).

Cf. A127917 (order of SL_2(p)), A047927.

[(NthPrime(n)-1)*NthPrime(n)*(NthPrime(n)^2-1): n in [1..100]]; // Vincenzo Librandi, Aug 15 2018

gl2psz[p_] := (p - 1) p (p^2 - 1); sqg = gl2psz/@Prime@Range[m]
Table[(Prime[n] - 1) Prime[n] (Prime[n]^2 - 1), {n, 30}] (* Vincenzo Librandi, Aug 15 2018 *)

a(n) = { my(p=prime(n)); (p-1)*p*(p^2-1) } \\ Joerg Arndt, Nov 23 2014

a(n) = (p-1)*p*(p^2-1) where p = prime(n).
a(n) = A127917(n)*(prime(n)-1).
Subsequence of A047927. - Michel Marcus, Nov 25 2014
Sum 1/a(n) = A382584. - R. J. Mathar, Mar 31 2025

A342239 Table read by upward antidiagonals: T(n,k) is the number of strings of length k over an n-letter alphabet that are bifix free; n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 0, 3, 2, 0, 4, 6, 4, 0, 5, 12, 18, 6, 0, 6, 20, 48, 48, 12, 0, 7, 30, 100, 180, 144, 20, 0, 8, 42, 180, 480, 720, 414, 40, 0, 9, 56, 294, 1050, 2400, 2832, 1242, 74, 0, 10, 72, 448, 2016, 6300, 11900, 11328, 3678, 148, 0, 11, 90, 648, 3528, 14112, 37620, 59500, 45132, 11034, 284, 0

Offset: 1

Peter Kagey, Mar 06 2021

			Table begins:
n\k | 1  2   3    4     5      6       7        8         9
----+------------------------------------------------------
  1 | 1  0   0    0     0      0       0        0         0
  2 | 2  2   4    6    12     20      40       74       148
  3 | 3  6  18   48   144    414    1242     3678     11034
  4 | 4 12  48  180   720   2832   11328    45132    180528
  5 | 5 20 100  480  2400  11900   59500   297020   1485100
  6 | 6 30 180 1050  6300  37620  225720  1353270   8119620
  7 | 7 42 294 2016 14112  98490  689430  4823994  33767958
  8 | 8 56 448 3528 28224 225344 1802752 14418488 115347904

Peter Kagey, Antidiagonals n = 1..100, flattened

Rows: A003000 (n=2), A019308 (n=3), A019309 (n=4).

Columns: A002378 (k=1), A045991 (k=2), A047927 (k=3).

T(n,0)    = n.
T(n,2k)   = n*T(n,2k-1) - T(n,k).
T(n,2k+1) = n*T(n,2k).

cp's OEIS Frontend

A245334 A factorial-like triangle read by rows: T(0,0) = 1; T(n+1,0) = T(n,0)+1; T(n+1,k+1) = T(n,0)*T(n,k), k=0..n.

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A212163 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the rhombic hexagonal square grid graph RH_(k,k).

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A212195 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the staggered hexagonal square grid graph SH_(k,k).

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A059238 Orders of the finite groups GL_2(K) when K is a finite field with q = A246655(n) elements.

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A244509 Order of GL_2(p), the general linear group over F_p, where p runs through the primes.

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A342239 Table read by upward antidiagonals: T(n,k) is the number of strings of length k over an n-letter alphabet that are bifix free; n >= 1, k >= 1.

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