A022166 -id:A022166 - cp's OEIS frontend

A344110 Triangle read by rows: T(n,k) = 2^(n*k), n >= 0, 0 <= k <= n.

1, 1, 2, 1, 4, 16, 1, 8, 64, 512, 1, 16, 256, 4096, 65536, 1, 32, 1024, 32768, 1048576, 33554432, 1, 64, 4096, 262144, 16777216, 1073741824, 68719476736, 1, 128, 16384, 2097152, 268435456, 34359738368, 4398046511104, 562949953421312

Offset: 0

Mohammad K. Azarian, May 10 2021

T(n, k) is the number of relations from an n-element set into a k-element set, n >= 0, 0 <= k <= n.

T(n,k) is the size of the right principal ideal generated by A where A is an n X n matrix over GF(2) having rank k. The right principal ideal of A contains precisely the matrices whose image is contained in the image of A. - Geoffrey Critzer, Sep 25 2022

			T(3,3) = number of relations from a 3-element set into a 3-element set=2^(3*3)=512.
Triangle begins:
   1
   1   2
   1   4      16
   1   8      64      512
   1  16     256     4096      65536
   1  32    1024    32768    1048576    33554432
   ...

Michael De Vlieger, Table of n, a(n) for n = 0..1325 (rows n = 0..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000079, A089072, A008292, A031971, A117401, A344260 (row sums).

Cf. A022166, A288853.

Mathematica
```
Table[2^(n*k), {n, 0, 10}, {k, 0, n}]
```

T(n,k) = 2^(n*k).

T(n,k) = Sum_{j=0..k} A288853(n,j)*A022166(n,j). - Geoffrey Critzer, Jan 02 2023

A158474 Triangle read by rows generated from (x-1)(x-2)(x-4)*...

Original entry on oeis.org

1, 1, -1, 1, -3, 2, 1, -7, 14, -8, 1, -15, 70, -120, 64, 1, -31, 310, -1240, 1984, -1024, 1, -63, 1302, -11160, 41664, -64512, 32768, 1, -127, 5334, -94488, 755904, -2731008, 4161536, -2097152, 1, -255, 21590, -777240, 12850368, -99486720, 353730560

Offset: 0

Gary W. Adamson, Mar 20 2009

Row sum of the unsigned triangle = A028361: (1, 2, 6, 30, 270, 4590, ...).
Right border of the unsigned triangle = A006125: (1, 1, 2, 8, 64, 1024, ...).
From Philippe Deléham, Mar 20 2009: (Start)
Unsigned triangle: A077957(n) DELTA A007179(n+1) = [1,0,2,0,4,0,8,0,16,0,32,0,...]DELTA[1,1,4,6,16,28,64,120,256,496,...], where DELTA is the operator defined in A084938.
Signed triangle: [1,0,2,0,4,0,8,0,16,0,...]DELTA[-1,-1,-4,-6,-16,-28,-64,...]. (End)

			First few rows of the triangle =
1;
1,  -1;
1,  -3,     2;
1,  -7,    14,     -8;
1, -15,    70,   -120,       64;
1, -31,   310,  -1240,     1984,    -1024;
1, -63,  1302, -11160,    41664,   -64512,     32768;
1,-127,  5334, -94488,   755904, -2731008,   4161536,  -2097152;
1,-255, 21590,-777240, 12850368,-99486720, 353730560,-534773760, 268435456;
...
Example: row 3 = x^3 - 7x^2 + 14x - 8 = (x-1)*(x-2)*(x-4).

Cf. A028361, A006125.

Cf. A157963, A135950. - R. J. Mathar, Mar 20 2009

A158474 := proc(n,k) mul(x-2^j,j=0..n-1) ; expand(%); coeftayl(%,x=0,n-k) ; end proc: # R. J. Mathar, Aug 27 2011

{{1}}~Join~Table[Reverse@ CoefficientList[Fold[#1 (x - #2) &, 1, 2^Range[0, n]], x], {n, 0, 7}] // Flatten (* Michael De Vlieger, Dec 22 2016 *)

T(n,k) = coefficient [x^(n-k)] of (x-1)*(x-2)*(x-4)*...*(x-2^(n-1)).
T(n,k) = (-1)^k*(Sum_{j=0..k} T(k,j)*2^((k-j)*n))/(Product_{i=1..k} (2^i-1)) for n >= 0 and k > 0, i.e., e.g.f. of col k > 0 is: (-1)^k*(Sum_{j=0..k} T(k,j)* exp(2^(k-j)*t))/(Product_{i=1..k} (2^i-1)). - Werner Schulte, Dec 18 2016
T(n,k)/T(k,k) = A022166(n,k) for 0 <= k <= n. - Werner Schulte, Dec 21 2016

A296548 Triangle read by rows: T(n,k) is the number of diagonalizable n X n matrices over GF(2) that have rank k, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 28, 28, 1, 1, 120, 560, 120, 1, 1, 496, 9920, 9920, 496, 1, 1, 2016, 166656, 714240, 166656, 2016, 1, 1, 8128, 2731008, 48377856, 48377856, 2731008, 8128, 1, 1, 32640, 44216320, 3183575040, 13158776832, 3183575040, 44216320, 32640, 1

Offset: 0

Geoffrey Critzer, Dec 15 2017

Equivalently, T(n,k) is the number of n X n matrices, P, over GF(2) with rank k, such that P^2 = P.

Equivalently, T(n,k) is the number of direct sum decompositions of the vector space GF(2)^n into exactly two subspaces U and W such that the dimension of U is k.

			Triangle T(n,k) begins:
  1;
  1,    1;
  1,    6,      1;
  1,   28,     28,      1;
  1,  120,    560,    120,      1;
  1,  496,   9920,   9920,    496,    1;
  1, 2016, 166656, 714240, 166656, 2016, 1;
  ...

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Cf. A132186 (row sums).

b:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, b(n-1, k-1)+2^k*b(n-1, k)))
    end:
T:= (n,k)-> 2^(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; g[n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[
    QFactorial[n, q]] /. q -> 2; Grid[Map[Select[#, # > 0 &] &,
  Table[g[n], {n, 0, nn}] CoefficientList[Series[Sum[(u z)^r/g[r] , {r, 0, nn}] Sum[z^r/g[r], {r, 0, nn}], {z, 0, nn}], {z, u}]]]

T(n,k)/A002884(n) is the coefficient of y^k*x^n in the expansion of Sum_{n>=0} x^n\A002884(n) * Sum_{n>=0} y*x^n\A002884(n).

T(n,k) = A002884(n)/(A002884(k)*A002884(n-k)) = A022166(n,k)*2^(k(n-k)).

A362824 Array read by antidiagonals: T(n,k) is the number of k-tuples of involutions on [n] that pairwise commute.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 4, 1, 1, 1, 8, 10, 10, 1, 1, 1, 16, 22, 52, 26, 1, 1, 1, 32, 46, 232, 196, 76, 1, 1, 1, 64, 94, 976, 1016, 1216, 232, 1, 1, 1, 128, 190, 4000, 4576, 12496, 5944, 764, 1, 1, 1, 256, 382, 16192, 19376, 111376, 73648, 42400, 2620, 1

Offset: 0

Andrew Howroyd, May 06 2023

Two involutions x,y on [n] commute if x*y = y*x.

			Array begins:
===========================================================
n/k| 0   1    2     3      4       5        6         7 ...
---+-------------------------------------------------------
0  | 1   1    1     1      1       1        1         1 ...
1  | 1   1    1     1      1       1        1         1 ...
2  | 1   2    4     8     16      32       64       128 ...
3  | 1   4   10    22     46      94      190       382 ...
4  | 1  10   52   232    976    4000    16192     65152 ...
5  | 1  26  196  1016   4576   19376    79696    323216 ...
6  | 1  76 1216 12496 111376  936976  7680016  62177296 ...
7  | 1 232 5944 73648 716416 6289312 52647904 430723168 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).

Columns k=0..3 are A000012, A000085, A362819, A362825.
Rows n=2..3 are A000079, A033484.
Main diagonal is A362823.
Cf. A022166, A362648.

\\ B(n,k) is A022166.
B(n,k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}
T(n,k)={if(n==0, 1, n!*polcoef(exp(sum(j=0, min(k,logint(n,2)), B(k,j)*x^(2^j)/2^j, O(x*x^n))), n))}

T(0,k) = T(1,k) = 1.

A034496 Sum of n-th powers of divisors of 8.

Original entry on oeis.org

4, 15, 85, 585, 4369, 33825, 266305, 2113665, 16843009, 134480385, 1074791425, 8594130945, 68736258049, 549822930945, 4398314962945, 35185445863425, 281479271743489, 2251816993685505, 18014467229220865

Offset: 0

N. J. A. Sloane

Conjecture: No primes in this sequence (checked for first 10000 terms). [Artur Jasinski, Sep 23 2008]

All terms are composite because a(n) = (1 + 2^n)*(1 + 4^n). [T. D. Noe, Apr 26 2010]

T. D. Noe, Table of n, a(n) for n = 0..200
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.

Cf. A006096, A022166.

[DivisorSigma(n,8): n in [0..20]]; // Vincenzo Librandi, Apr 17 2014

Total[#^Range[0, 20]&/@Divisors[8]] (* Vincenzo Librandi, Apr 17 2014 *)
DivisorSigma[Range[0,20],8] (* Harvey P. Dale, May 16 2020 *)

a(n)=sigma(8, n) \\ Charles R Greathouse IV, May 16 2011

[sigma(8,n) for n in range(0,19)] # Zerinvary Lajos, Jun 04 2009

G.f.: (4 - 45*x + 140*x^2 - 120*x^3)/((1 - 8*x)*(1 - 4*x)*(1 - 2*x)*(1 - x)). [Bruno Berselli, Apr 17 2014]

a(n) = (2^(4*n) - 1)/( 2^n - 1) = 1 + 2^n + 4^n + 8^n. Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 15*x + 155*x^2 + 1395*x^3 + ... is the o.g.f. for the 3rd subdiagonal of triangle A022166, essentially A006096. - Peter Bala, Apr 07 2015

A286331 Triangle read by rows: T(n,k) is the number of n X n matrices of rank k over F_2.

Original entry on oeis.org

1, 1, 1, 1, 9, 6, 1, 49, 294, 168, 1, 225, 7350, 37800, 20160, 1, 961, 144150, 4036200, 19373760, 9999360, 1, 3969, 2542806, 326932200, 8543828160, 39687459840, 20158709760, 1, 16129, 42677334, 23435953128, 2812314375360, 71124337751040, 325139829719040, 163849992929280

Offset: 0

Geoffrey Critzer, May 07 2017

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   9,      6;
  1,  49,    294,     168;
  1, 225,   7350,   37800,    20160;
  1, 961, 144150, 4036200, 19373760, 9999360;
  ...
T(2,1) = 9 because there are 9, 2 X 2 matrices in F_2 that have rank 1: {{0, 0}, {0, 1}}, {{0, 0}, {1, 0}}, {{0, 0}, {1, 1}}, {{0, 1}, {0, 0}}, {{0, 1}, {0, 1}}, {{1, 0},  {0, 0}}, {{1, 0}, {1, 0}}, {{1,1}, {0, 0}}, {{1, 1}, {1, 1}}.

Robert Israel, Table of n, a(n) for n = 0..1710 (rows 0 to 57, flattened)
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Wikipedia, q-binomial

Main diagonal is A002884.
Column for k = 1 is A060867.
Row sums are A002416.

T:= (n,k) -> mul((2^n-2^j)^2/(2^k-2^j),j=0..k-1):
seq(seq(T(n,k),k=0..n),n=0..10); # Robert Israel, May 15 2017

q = 2; Table[Table[Product[(q^n - q^i)^2/(q^k - q^i), {i, 0, k - 1}], {k, 0, n}], {n, 0, 6}] // Grid

T(n,k) = Product_{j=0..k-1} (2^n - 2^j)^2/(2^k - 2^j) = A022166(n,k) * Product_{j=0..k-1} (2^n - 2^j).

A020514 a(n) = 1^n + 2^n + 4^n + 8^n + 16^n.

Original entry on oeis.org

5, 31, 341, 4681, 69905, 1082401, 17043521, 270549121, 4311810305, 68853957121, 1100586419201, 17600780175361, 281543712968705, 4504149450301441, 72061992352890881, 1152956690052710401, 18447025552981295105, 295150156996346511361, 4722384497336874434561

Offset: 0

Simon Plouffe and N. J. A. Sloane

5th cyclotomic polynomial evaluated at 2^n.

T. D. Noe, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (31,-310,1240,-1984,1024).

Cf. A082771, A006097, A022166.

with(numtheory,cyclotomic):seq(cyclotomic(5,2^i),i=0..24);

With[{c=2^Range[0,4]},Table[Total[c^n],{n,0,20}]] (* Harvey P. Dale, May 27 2012 *)

a(n)=1^n+2^n+4^n+8^n+16^n \\ Charles R Greathouse IV, Oct 07 2015

[sigma(16,n)for n in range(0,16)] # Zerinvary Lajos, Jun 04 2009

G.f.: 1/(1-x)+1/(1-2*x)+1/(1-4*x)+1/(1-8*x)+1/(1-16*x). - Philippe Deléham, Apr 06 2013
E.g.f.: exp(x) + exp(2*x) + exp(4*x) + exp(8*x) + exp(16*x). - Philippe Deléham, Apr 06 2013
a(n) = 31*a(n-1) - 310*a(n-2) + 1240*a(n-3) - 1984*a(n-4) + 1024*a(n-5) with a(0) = 5, a(1) = 31, a(2) = 341, a(3) = 4681, a(4) = 69905. - Philippe Deléham, Apr 06 2013
a(n) = (2^(5*n) - 1)/( 2^n - 1). Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 31*x + 651*x^2 + 11811*x^3 + ... is the o.g.f. for the 4th subdiagonal of triangle A022166, essentially A006097. - Peter Bala, Apr 07 2015

A020516 Sum of n-th powers of divisors of 64.

Original entry on oeis.org

7, 127, 5461, 299593, 17895697, 1108378657, 69810262081, 4432676798593, 282578800148737, 18049651735527937, 1154048505100108801, 73823022692637345793, 4723519685917965029377, 302268352895954163081217

Offset: 0

Simon Plouffe

7th cyclotomic polynomial evaluated at powers of 2.

T. D. Noe, Table of n, a(n) for n = 0..200

Cf. A022166, A022189.

[&+[Divisors(64)[i]^n: i in [1..7]]: n in [0..15]]; // Vincenzo Librandi, Apr 17 2014

with(numtheory,cyclotomic):seq(cyclotomic(7,2^i),i=0..24);

Total[#^Range[0,15]&/@Divisors[64]]  (* Harvey P. Dale, Mar 21 2011 *)

a(n) = polcyclo(7, 2^n); \\ Michel Marcus, Nov 13 2016

G.f.: (-7+762 x-26670 x^2+377952 x^3-2267712 x^4+5462016 x^5-4161536 x^6)/(-1+127 x-5334 x^2+94488 x^3-755904 x^4+2731008 x^5-4161536 x^6+2097152 x^7). - Harvey P. Dale, Mar 21 2011

a(n) = (2^(7*n) - 1)/( 2^n - 1). Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 127*x + 10795*x^2 + ... is the o.g.f. for the 6th subdiagonal of triangle A022166, essentially A022189. - Peter Bala, Apr 07 2015

A034665 Sum of n-th powers of divisors of 32.

Original entry on oeis.org

6, 63, 1365, 37449, 1118481, 34636833, 1090785345, 34630287489, 1103823438081, 35253226045953, 1127000493261825, 36046397799139329, 1153203048319815681, 36897992296869404673, 1180663682709764194305

Offset: 0

N. J. A. Sloane

T. D. Noe, Table of n, a(n) for n = 0..200
Quynh Nguyen, Jean Pedersen, and Hien T. Vu, New Integer Sequences Arising From 3-Period Folding Numbers, Vol. 19 (2016), Article 16.3.1. See Table 1.
Index entries for linear recurrences with constant coefficients, signature (63,-1302,11160,-41664,64512,-32768).

Cf. A006110, A022166.

[DivisorSigma(n,32): n in [0..15]]; // Vincenzo Librandi, Apr 17 2014

Total[#^Range[0, 15]&/@Divisors[32]] (* Vincenzo Librandi, Apr 17 2014 *)
LinearRecurrence[{63,-1302,11160,-41664,64512,-32768},{6,63,1365,37449,1118481,34636833},20] (* Harvey P. Dale, Jan 10 2015 *)

a(n)=(64^n-1)/(2^n-1) \\ Charles R Greathouse IV, Oct 07 2015

print([1+2**n+4**n+8**n+16**n+32**n for n in range(15)]) # Karl V. Keller, Jr., Feb 02 2021

[sigma(32,n)for n in range(0,15)] # Zerinvary Lajos, Jun 04 2009

G.f.: -3*(21504*x^5-27776*x^4+11160*x^3-1736*x^2+105*x-2) / ((x-1)*(2*x-1)*(4*x-1)*(8*x-1)*(16*x-1)*(32*x-1)). - Colin Barker, Apr 20 2014
a(n) = (2^(6*n) - 1)/( 2^n - 1). Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 63*x + 2667*x^2 + 97155*x^3 + ... is the o.g.f. for the 5th subdiagonal of triangle A022166, essentially A006110. - Peter Bala, Apr 07 2015
a(n) = 1 + 2^n + 4^n + 8^n + 16^n + 32^n for n>=0. - Karl V. Keller, Jr., Feb 02 2021

A157783 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (3^(i-1)-x) in row n, column k, 0 <= k <= n.

Original entry on oeis.org

1, 1, -1, 3, -4, 1, 27, -39, 13, -1, 729, -1080, 390, -40, 1, 59049, -88209, 32670, -3630, 121, -1, 14348907, -21493836, 8027019, -914760, 33033, -364, 1, 10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093

Offset: 0

Roger L. Bagula, Mar 06 2009

Row sums except n=0 are zero.

Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=3)=[1,2,9,24,81,234,729,2160,6561,...] DELTA [ -1,0,-3,0,-9,0,-27,0,-81,0,-243,0,...] where DELTA is the operator defined in A084938; see A122006 and A000244. - Philippe Deléham, Mar 09 2009

			Triangle begins
  1;
  1, -1;
  3, -4, 1;
  27, -39, 13, -1;
  729, -1080, 390, -40, 1;
  59049, -88209, 32670, -3630, 121, -1;
  14348907, -21493836, 8027019, -914760, 33033, -364, 1;
  10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093, -1;
  22876792454961, -34309958505840, 12860351387820, -1481851188720, 55340738838, -677572560, 2688780, -3280, 1;
Row n=3 is 27 - 39*x + 13*x^2 - x^3.

Cf. A157832, A135950, A022166, A047656 (column k=1), A003462 (subdiagonal k=n-1), A203243 (subdiagonal k=n-2).

A157783 := proc(n,k)
    product( 3^(i-1)-x,i=1..n) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

Clear[f, q, M, n, m];
q = 3;
f[k_, m_] := If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[k == n && m > 1, 1, 0]]]];
M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}];
Table[M[n], {n, 1, 10}];
Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]];
a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]];
Flatten[a]

cp's OEIS Frontend

A344110 Triangle read by rows: T(n,k) = 2^(n*k), n >= 0, 0 <= k <= n.

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A158474 Triangle read by rows generated from (x-1)*(x-2)*(x-4)*...

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A296548 Triangle read by rows: T(n,k) is the number of diagonalizable n X n matrices over GF(2) that have rank k, n >= 0, 0 <= k <= n.

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A362824 Array read by antidiagonals: T(n,k) is the number of k-tuples of involutions on [n] that pairwise commute.

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A034496 Sum of n-th powers of divisors of 8.

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A286331 Triangle read by rows: T(n,k) is the number of n X n matrices of rank k over F_2.

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A020514 a(n) = 1^n + 2^n + 4^n + 8^n + 16^n.

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A158474 Triangle read by rows generated from (x-1)(x-2)(x-4)*...