A006951 -id:A006951 - cp's OEIS frontend

A264687 a(n) = 2^n - A006951(n).

0, 1, 1, 2, 2, 5, 4, 11, 10, 22, 22, 50, 43, 104, 100, 209, 206, 446, 418, 914, 886, 1838, 1825, 3800, 3664, 7696, 7591, 15470, 15370, 31466, 30910, 63398, 62866, 127154, 126742, 256559, 254156, 515168, 513004, 1032104, 1029997, 2073626, 2063866, 4156118

Offset: 0

Vaclav Kotesovec, Nov 21 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A006951, A304082.

nmax = 60; 2^Range[0, nmax] - CoefficientList[Series[Product[(1 - x^k)/(1 - 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (1+sqrt(2) + (-1)^n*(1-sqrt(2))) * 2^(n/2 - 1).

A133494 Diagonal of the array of iterated differences of A047848.

Original entry on oeis.org

1, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883

Offset: 0

Paul Barry, Paul Curtz, Dec 23 2007

a(n) is the number of ways to choose a composition C, and then choose a composition of each part of C. - Geoffrey Critzer, Mar 19 2012
a(n) is the top left entry of the n-th power of the 3 X 3 matrix [1, 1, 1; 1, 1, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014
a(n) is the reptend length of 1/3^(n+1) in decimal. - Jianing Song, Nov 14 2018
Also the number of pairs of integer compositions, the first summing to n and the second with sum equal to the length of the first. If an integer composition is regarded as an arrow from sum to length, these are composable pairs, and the obvious composition operation founds a category of integer compositions. For example, we have (2,1,1,4) . (1,2,1) . (1,2) = (2,6), where dots represent the composition operation. The version without empty compositions is A000244. Composable triples are counted by 1 followed by A000302. The unordered version is A022811. - Gus Wiseman, Jul 14 2022

			From _Gus Wiseman_, Jul 15 2020: (Start)
The a(0) = 1 through a(3) = 9 ways to choose a composition of each part of a composition:
  ()  (1)  (2)      (3)
           (1,1)    (1,2)
           (1),(1)  (2,1)
                    (1,1,1)
                    (1),(2)
                    (2),(1)
                    (1),(1,1)
                    (1,1),(1)
                    (1),(1),(1)
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Bishal Deb, Cyclic sieving phenomena via combinatorics of continued fractions, arXiv:2508.13709 [math.CO], 2025. See p. 42.
Index entries for linear recurrences with constant coefficients, signature (3).

The strict version is A336139.
Splittings of partitions are A323583.
Multiset partitions of partitions are A001970.
Partitions of each part of a partition are A063834.
Compositions of each part of a partition are A075900.
Strict partitions of each part of a strict partition are A279785.
Compositions of each part of a strict partition are A304961.
Strict compositions of each part of a composition are A307068.
Compositions of each part of a strict composition are A336127.
Cf. A000012, A000244, A000302, A001906, A006951, A011782, A022811, A071919.
Cf. A072706, A078008, A112467, A182105, A316245, A336128, A336135.
Cf. A339006, A355382, A355384, A355387.

[n eq 0 select 1 else 3^(n-1): n in [0..30]]; // G. C. Greubel, Nov 20 2023

a:= n-> ceil(3^(n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 26 2020

CoefficientList[Series[(1 - 2 x)/(1 - 3 x), {x, 0, 50}], x] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)
Join[{1}, 3^(Range[0, 30])] (* G. C. Greubel, Nov 20 2023 *)

a(n)=max(1,3^(n-1)) \\ Charles R Greathouse IV, Jul 07 2011

Vec((1-2*x)/(1-3*x) + O(x^100)) \\  Altug Alkan, Oct 30 2015

[(3^n + 2*int(n==0))//3 for n in range(31)] # G. C. Greubel, Nov 20 2023

Binomial transform of A078008. - Paul Curtz, Aug 04 2008
From R. J. Mathar, Nov 11 2008: (Start)
G.f.: (1 - 2*x)/(1 - 3*x).
a(n) = A000244(n-1), n > 0. (End)
From Philippe Deléham, Nov 13 2008: (Start)
a(n) = Sum_{k=0..n} A112467(n,k)*2^k.
a(n) = Sum_{k=0..n} A071919(n,k)*2^k. (End)
Let A(x) be the g.f. Then B(x) = x*A(x) satisfies B(x/(1-x)) = x/(1 - 2*B(x)). - Vladimir Kruchinin, Dec 05 2011
G.f.: 1/(1 - (Sum_{k>=1} (x/(1 - x))^k)). - Joerg Arndt, Sep 30 2012
For n > 0, a(n) = 2*(Sum_{k=0..n-1} a(k)) - 1 = 3^(n-1). - J. Conrad, Oct 29 2015
G.f.: 1 + x/(1 + x)*(1 + 4*x/(1 + 4*x)*(1 + 7*x/(1 + 7*x)*(1 + 10*x/(1 + 10*x)*(1 + .... - Peter Bala, May 27 2017
Invert transform of A011782(n) = 2^(n-1). Second invert transform of A000012. - Gus Wiseman, Jul 19 2020
a(n) = ceiling(3^(n-1)). - Alois P. Heinz, Jul 26 2020
From Elmo R. Oliveira, Mar 31 2025: (Start)
E.g.f.: (2 + exp(3*x))/3.
a(n) = 3*a(n-1) for n > 1. (End)

Definition clarified by R. J. Mathar, Nov 11 2008

A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 13, 19, 26, 36, 43, 64, 77, 102, 129, 169, 205, 268, 323, 413, 504, 629, 751, 947, 1131, 1384, 1661, 2024, 2393, 2919, 3442, 4136, 4884, 5834, 6836, 8162, 9531, 11262, 13155, 15493, 17981, 21138, 24472, 28571, 33066, 38475, 44305

Offset: 0

David S. Newman, Nov 21 2004

nonn

From Gus Wiseman, Jan 21 2019: (Start)
Also the number of semistandard Young tableaux where the rows are constant and the entries sum to n. For example, the a(8) = 19 tableaux are:
  8   44   2222   11111111
.
  1   2   11   3   111   22   1111   11   11111   1111   111111
  7   6   6    5   5     4    4      33   3       22     2
.
  1   1   11   111
  2   3   2    2
  5   4   4    3
(End)

			a(5) = 6 because, of the 7 unrestricted partitions of 5, only one, 2 + 2 + 1, has a decreasing sequence of frequencies. Two is used twice, but 1 is used only once.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A100881, A100882, A100884.

Cf. A000085, A000219, A003293, A006951, A100471, A323582.

b:= proc(n, i, t) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i=1, `if`(n>=t, 1, 0), `if`(i=0, 0, b(n, i-1, t)+
       add(b(n-i*j, i-1, j), j=t..floor(n/i))))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 03 2014

b[n_, i_, t_] := b[n, i, t] = If[n<0, 0, If[n == 0, 1, If[i == 1, If[n >= t, 1, 0], If[i == 0, 0, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t, Floor[n/i]}]]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Length/@Split[#]]&]],{n,20}] (* Gus Wiseman, Jan 21 2019 *)

More terms from Vladeta Jovovic, Nov 23 2004

A075900 Expansion of g.f.: Product_{n>0} 1/(1 - 2^(n-1)*x^n).

Original entry on oeis.org

1, 1, 3, 7, 19, 43, 115, 259, 659, 1523, 3731, 8531, 20883, 47379, 113043, 259219, 609683, 1385363, 3245459, 7344531, 17028499, 38579603, 88585619, 199845267, 457864595, 1028904339, 2339763603, 5256820115, 11896157587, 26626389395

Offset: 0

N. J. A. Sloane, Oct 15 2002

nonn

Number of compositions of partitions of n. a(3) = 7: 3, 21, 12, 111, 2|1, 11|1, 1|1|1. - Alois P. Heinz, Sep 16 2019
Also the number of ways to split an integer composition of n into consecutive subsequences with weakly decreasing (or increasing) sums. - Gus Wiseman, Jul 13 2020
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1, g(n) = 2^(n-1). - Seiichi Manyama, Aug 22 2020

			From _Gus Wiseman_, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 19 splittings:
  ()  (1)  (2)      (3)          (4)
           (1,1)    (1,2)        (1,3)
           (1),(1)  (2,1)        (2,2)
                    (1,1,1)      (3,1)
                    (2),(1)      (1,1,2)
                    (1,1),(1)    (1,2,1)
                    (1),(1),(1)  (2,1,1)
                                 (2),(2)
                                 (3),(1)
                                 (1,1,1,1)
                                 (1,1),(2)
                                 (1,2),(1)
                                 (2),(1,1)
                                 (2,1),(1)
                                 (1,1),(1,1)
                                 (1,1,1),(1)
                                 (2),(1),(1)
                                 (1,1),(1),(1)
                                 (1),(1),(1),(1)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..3180 (terms 0..1000 from Alois P. Heinz)
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

Row sums of A327549.
The strict case is A304961.
Partitions of partitions are A001970.
Splittings with equal sums are A074854.
Splittings of compositions are A133494.
Splittings of partitions are A323583.
Splittings with distinct sums are A336127.
Starting with a reversed partition gives A316245.
Starting with a partition instead of composition gives A336136.
Cf. A006951, A063834, A300579, A317715, A319794, A323582, A327548, A336135.

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(&*[1-2^(j-1)*x^j: j in [1..m+2]]) )); // G. C. Greubel, Jan 25 2024

oo := 101; t1 := mul(1/(1-x^n/2),n=1..oo): t2 := series(t1,x,oo-1): t3 := seriestolist(t2): A075900 := n->2^n*t3[n+1];
with(combinat); A075900 := proc(n) local i,t1,t2,t3; t1 := partition(n); t2 := 0; for i from 1 to nops(t1) do t3 := t1[i]; t2 := t2+2^(n-nops(t3)); od: t2; end;

b[n_]:= b[n]= Sum[d*2^(n - n/d), {d, Divisors[n]}];
a[0]= 1; a[n_]:= a[n]= 1/n*Sum[b[k]*a[n-k], {k,n}];
Table[a[n], {n,0,30}] (* Jean-François Alcover, Mar 20 2014, after Vladeta Jovovic, fixed by Vaclav Kotesovec, Mar 08 2018 *)

s(m,n):=if nVladimir Kruchinin, Sep 06 2014 */

{a(n)=polcoeff(prod(k=1,n,1/(1-2^(k-1)*x^k+x*O(x^n))),n)} \\ Paul D. Hanna, Jan 13 2013

{a(n)=polcoeff(exp(sum(k=1,n+1,x^k/(k*(1-2^k*x^k)+x*O(x^n)))),n)} \\ Paul D. Hanna, Jan 13 2013

m=80;
def A075900_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( 1/product(1-2^(j-1)*x^j for j in range(1,m+1)) ).list()
A075900_list(m) # G. C. Greubel, Jan 25 2024

a(n) = Sum_{ partitions n = c_1 + ... + c_k } 2^(n-k). If p(n, m) = number of partitions of n into m parts, a(n) = Sum_{m=1..n} p(n, m)*2^(n-m).
G.f.: Sum_{n>=0} (a(n)/2^n)*x^n = Product_{n>0} 1/(1-x^n/2). - Vladeta Jovovic, Feb 11 2003
a(n) = 1/n*Sum_{k=1..n} A080267(k)*a(n-k). - Vladeta Jovovic, Feb 11 2003
G.f.: exp( Sum_{n>=1} x^n / (n*(1 - 2^n*x^n)) ). - Paul D. Hanna, Jan 13 2013
a(n) = s(1,n), a(0)=1, where s(m,n) = Sum_{k=m..n/2} 2^(k-1)*s(k, n-k)  + 2^(n-1), s(n,n) = 2^(n-1), s(m,n)=0, m>. - Vladimir Kruchinin, Sep 06 2014
a(n) ~ 2^(n-2) * (Pi^2 - 6*log(2)^2)^(1/4) * exp(sqrt((Pi^2 - 6*log(2)^2)*n/3)) / (3^(1/4) * sqrt(Pi) * n^(3/4)). - Vaclav Kotesovec, Mar 09 2018

More terms from Vladeta Jovovic, Feb 11 2003

A070933 Expansion of Product_{k>=1} 1/(1 - 2*t^k).

Original entry on oeis.org

1, 2, 6, 14, 34, 74, 166, 350, 746, 1546, 3206, 6550, 13386, 27114, 54894, 110630, 222794, 447538, 898574, 1801590, 3610930, 7231858, 14480654, 28983246, 58003250, 116054034, 232186518, 464475166, 929116402, 1858449178, 3717247638, 7434950062, 14870628026, 29742206138, 59485920374, 118973809798, 237950730522, 475905520474

Offset: 0

Sharon Sela (sharonsela(AT)hotmail.com), May 21 2002

nonn

See A083355 for a similar formula. - Thomas Wieder, May 07 2008
Partitions of n into 2 sorts of parts: the parts are unordered, but not the sorts; see example and formula by Wieder. - Joerg Arndt, Apr 28 2013
Convolution inverse of A070877. - George Beck, Dec 02 2018
Number of conjugacy classes of n X n matrices over GF(2).  Cf. Morrison link, section 2.9. - Geoffrey Critzer, May 26 2021

			From _Joerg Arndt_, Apr 28 2013: (Start)
There are a(3)=14 partitions of 3 with 2 ordered sorts. Here p:s stands for "part p of sort s":
01:  [ 1:0  1:0  1:0  ]
02:  [ 1:0  1:0  1:1  ]
03:  [ 1:0  1:1  1:0  ]
04:  [ 1:0  1:1  1:1  ]
05:  [ 1:1  1:0  1:0  ]
06:  [ 1:1  1:0  1:1  ]
07:  [ 1:1  1:1  1:0  ]
08:  [ 1:1  1:1  1:1  ]
09:  [ 2:0  1:0  ]
10:  [ 2:0  1:1  ]
11:  [ 2:1  1:0  ]
12:  [ 2:1  1:1  ]
13:  [ 3:0  ]
14:  [ 3:1  ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from T. D. Noe)
Dragomir Z. Djokovic, Poincaré series of some pure and mixed trace algebras of two generic matrices, Journal of Algebra, Vol. 309, No. 2 (2007), 654-671, arXiv:math/0609262.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Igor Pak, Greta Panova, and Damir Yeliussizov, On the largest Kronecker and Littlewood-Richardson coefficients, arXiv:1804.04693 [math.CO], 2018.
N. J. A. Sloane, Transforms.

Cf. A006951, A000041, A070877.
Cf. A083355.
Column k=2 of A246935.
Cf. A048651.
Row sums of A256193.
Antidiagonal sums of A322210.

m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-2*x^k): k in [1..m]]) )); // G. C. Greubel, Oct 31 2018

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 07 2014

CoefficientList[ Series[ Product[1 / (1 - 2t^k), {k, 1, 35}], {t, 0, 35}], t]
CoefficientList[Series[E^Sum[2^k*x^k / (k*(1-x^k)), {k,1,30}],{x,0,30}],x] (* Vaclav Kotesovec, Sep 09 2014 *)
(O[x]^20 - 1/QPochhammer[2,x])[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

N=66; q='q+O('q^N); Vec(1/sum(n=0, N, (-2)^n*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) )) \\ Joerg Arndt, Mar 09 2014

a(n) = (1/n)*Sum_{k=1..n} A054598(k)*a(n-k). - Vladeta Jovovic, Nov 23 2002
a(n) is asymptotic to c*2^n where c=3.46253527447396564949732... - Benoit Cloitre, Oct 26 2003. Right value of this constant is c = 1/A048651 = 3.46274661945506361153795734292443116454075790290443839132935303175891543974042... . - Vaclav Kotesovec, Sep 09 2014
Euler transform of A000031(n). - Vladeta Jovovic, Jun 23 2004
a(n) = Sum_{k=1..n} p(n,k)*A000079(k) where p(n,k) = number of integer partitions of n into k parts. - Thomas Wieder, May 07 2008
a(n) = S(n,1), where S(n,m) = 2 + Sum_{k=m..floor(n/2)} 2*S(n-k,k), S(n,n)=2, S(0,m)=1, S(n,m)=0 for n < m. - Vladimir Kruchinin, Sep 07 2014
a(n) = Sum_{lambda,mu,nu} (c^{lambda}{mu,nu})^2, where lambda ranges over all partitions of n, mu and nu range over all partitions satisfying |mu| + |nu| = n, and c^{lambda}{mu,nu} denotes a Littlewood-Richardson coefficient. - Richard Stanley, Nov 16 2014
G.f.: Sum_{i>=0} 2^i*x^i/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018
G.f.: Product_{j>=1} Product_{i>=1} 1/(1-x^(i*j))^A001037(j) given in Morrison link section 2.9. - Geoffrey Critzer, May 26 2021

Edited and extended by Robert G. Wilson v, May 25 2002

A304961 Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).

Original entry on oeis.org

1, 1, 2, 6, 12, 32, 72, 176, 384, 960, 2112, 4992, 11264, 26112, 58368, 136192, 301056, 688128, 1548288, 3489792, 7766016, 17596416, 38993920, 87293952, 194248704, 432537600, 957349888, 2132803584, 4699717632, 10406068224, 23001563136, 50683969536, 111434268672, 245819768832

Offset: 0

Ilya Gutkovskiy, May 22 2018

nonn

Number of compositions of partitions of n into distinct parts. a(3) = 6: 3, 21, 12, 111, 2|1, 11|1. - Alois P. Heinz, Sep 16 2019
Also the number of ways to split a composition of n into contiguous subsequences with strictly decreasing sums. - Gus Wiseman, Jul 13 2020
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = (-1) * 2^(n-1). - Seiichi Manyama, Aug 22 2020

			From _Gus Wiseman_, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 12 splittings:
  ()  (1)  (2)    (3)        (4)
           (1,1)  (1,2)      (1,3)
                  (2,1)      (2,2)
                  (1,1,1)    (3,1)
                  (2),(1)    (1,1,2)
                  (1,1),(1)  (1,2,1)
                             (2,1,1)
                             (3),(1)
                             (1,1,1,1)
                             (1,2),(1)
                             (2,1),(1)
                             (1,1,1),(1)
(End)

Cf. A000009, A011782, A022629, A098407, A102866, A266964, A271619, A279785.
The non-strict version is A075900.
Starting with a reversed partition gives A323583.
Starting with a partition gives A336134.
Partitions of partitions are A001970.
Splittings with equal sums are A074854.
Splittings of compositions are A133494.
Splittings with distinct sums are A336127.
Cf. A006951, A063834, A316245, A317715, A319794, A323582, A336135, A336136.

nmax = 33; CoefficientList[Series[Product[(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]

N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+2^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020

G.f.: Product_{k>=1} (1 + A011782(k)*x^k).

a(n) ~ 2^n * exp(2*sqrt(-polylog(2, -1/2)*n)) * (-polylog(2, -1/2))^(1/4) / (sqrt(6*Pi) * n^(3/4)). - Vaclav Kotesovec, Sep 19 2019

A323583 Number of ways to split an integer partition of n into consecutive subsequences.

Original entry on oeis.org

1, 1, 3, 7, 17, 37, 83, 175, 373, 773, 1603, 3275, 6693, 13557, 27447, 55315, 111397, 223769, 449287, 900795, 1805465, 3615929, 7240327, 14491623, 29001625, 58027017, 116093259, 232237583, 464558201, 929224589, 1858623819, 3717475031, 7435314013, 14871103069

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

			The a(3) = 7 ways to split an integer partition of 3 into consecutive subsequences are (3), (21), (2)(1), (111), (11)(1), (1)(11), (1)(1)(1).

Cf. A006951, A070933, A100883, A279784, A279786, A323433, A323582.

b:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
    end:
a:= n-> ceil(b(n$2)):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 01 2023

Table[Sum[2^(Length[ptn]-1),{ptn,IntegerPartitions[n]}],{n,40}]
(* Second program: *)
(1/2) CoefficientList[1 - 1/QPochhammer[2, x] + O[x]^100 , x] (* Jean-François Alcover, Jan 02 2022, after Vladimir Reshetnikov in A070933 *)

a(n) = A070933(n)/2.
O.g.f.: (1/2)*Product_{n >= 1} 1/(1 - 2*x^n).
G.f.: 1 + Sum_{k>=1} 2^(k - 1) * x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 28 2020

A006952 Number of conjugacy classes in GL(n,3).

Original entry on oeis.org

1, 2, 8, 24, 78, 232, 720, 2152, 6528, 19578, 58944, 176808, 531128, 1593288, 4781952, 14345792, 43043622, 129130584, 387411144, 1162232520, 3486755688, 10460266224, 31380972784, 94142915640, 282429275616, 847287817866, 2541865038832, 7625595108432

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. D. Smith, personal communication.

Alois P. Heinz, Table of n, a(n) for n = 0..700
W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. Journal, 27 (1960) 91-94.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 162
I. G. Macdonald, Numbers of conjugacy classes in some finite classical groups, Bulletin of the Australian Mathematical Society, vol.23, no.01, pp.23-48, (February-1981).

Cf. A006951, A049314, A049315, A049316, A304082.

/* The program does not work for n>12: */ [1] cat [NumberOfClasses(GL(n, 3)) : n in [1..12]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by Vincenzo Librandi, Jan 23 2013

with(numtheory):
b:= n-> add(phi(d)*3^(n/d), d=divisors(n))/n-1:
a:= proc(n) option remember; `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2012

b[n_] := Sum[EulerPhi[d]*3^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] =  If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

N=66; x='x+O('x^N);
gf=prod(n=1, N, (1-x^n)/(1-3*x^n)  );
v=Vec(gf)
/* Joerg Arndt, Jan 02 2013 */

G.f.: Product_{n>=1} (1-x^n)/(1-3*x^n). - Joerg Arndt, Jan 02 2013
The number a(n) of conjugacy classes in the group GL(n, q) is the coefficient of t^n in Product_{k>=1} (1-t^k)/(1-q*t^k). - Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001
a(n) ~ 3^n - (1+sqrt(3) + (-1)^n*(1-sqrt(3))) * 3^(n/2) / 4. - Vaclav Kotesovec, May 06 2018
G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(3^(k/d) - 1) ) * x^k/k). - Ilya Gutkovskiy, Sep 27 2018

More terms from Alois P. Heinz, Nov 03 2012

A049314 The number k(GL(n,q)) of conjugacy classes in GL(n,q), q=4.

Original entry on oeis.org

1, 3, 15, 60, 252, 1005, 4080, 16305, 65460, 261828, 1048260, 4192980, 16775955, 67103520, 268430160, 1073720415, 4294945932, 17179782540, 68719391100, 274877559420, 1099511281260, 4398045120300, 17592184654365, 70368738597600, 281474971147680

Offset: 0

Vladeta Jovovic

nonn

Bound: k(GL(n,q))

Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Alois P. Heinz, Table of n, a(n) for n = 0..500
W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. Journal, 27 (1960) 91-94.

Cf. A006951, A006952, A049315, A049316.

/* The program does not work for n>9: */ [1] cat [NumberOfClasses(GL(n,4)) : n in [1..8]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by Vincenzo Librandi, Jan 23 2013

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*4^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2012

b[n_] := Sum[EulerPhi[d]*4^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

x='x+O('x^30); Vec(prod(n=1, 30, (1-x^n)/(1-4*x^n))) \\ Altug Alkan, Sep 27 2018

The number a(n) of conjugacy classes in the group GL(n, q) is the coefficient of t^n in the infinite product: product k=1, 2, ... (1-t^k)/(1-qt^k) - Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001.

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(4^(k/d) - 1) ) * x^k/k). - Ilya Gutkovskiy, Sep 27 2018

A049315 The number k(GL(n,q)) of conjugacy classes in GL(n,q), q=5.

Original entry on oeis.org

1, 4, 24, 120, 620, 3096, 15600, 77976, 390480, 1952380, 9764880, 48824280, 244136904, 1220683800, 6103496400, 30517481424, 152587794020, 762938966520, 3814696782120, 19073483892120, 95367429207720, 476837146020720, 2384185778835696, 11920928894086200

Offset: 0

Vladeta Jovovic

nonn

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Alois P. Heinz, Table of n, a(n) for n = 0..450
W. Feit and N. J. Fine, Pairs of commuting matrices over a finite field, Duke Math. Journal, 27 (1960) 91-94.

Cf. A006951, A006952, A049314, A049316.

/* The program does not work for n>8: */ [1] cat [NumberOfClasses(GL(n,5)) : n in [1..8]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006; edited by Vincenzo Librandi, Jan 23 2013

with(numtheory):
b:= proc(n) b(n):= add(phi(d)*5^(n/d), d=divisors(n))/n-1 end:
a:= proc(n) a(n):= `if`(n=0, 1,
       add(add(d*b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 03 2012

b[n_] := Sum[EulerPhi[d]*5^(n/d), {d, Divisors[n]}]/n-1; a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*b[d], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)

x='x+O('x^30); Vec(prod(n=1, 30, (1-x^n)/(1-5*x^n))) \\ Altug Alkan, Sep 27 2018

The number a(n) of conjugacy classes in the group GL(n, q) is the coefficient of t^n in the infinite product: product k=1, 2, ... (1-t^k)/(1-qt^k) - Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001.

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} d*(5^(k/d) - 1) ) * x^k/k). - Ilya Gutkovskiy, Sep 27 2018

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A264687 a(n) = 2^n - A006951(n).

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A133494 Diagonal of the array of iterated differences of A047848.

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A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.

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A075900 Expansion of g.f.: Product_{n>0} 1/(1 - 2^(n-1)*x^n).

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A070933 Expansion of Product_{k>=1} 1/(1 - 2*t^k).

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A304961 Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).

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