A006383 -id:A006383 - cp's OEIS frontend

A000601 Expansion of 1/((1-x)^2(1-x^2)(1-x^3)).

1, 2, 4, 7, 11, 16, 23, 31, 41, 53, 67, 83, 102, 123, 147, 174, 204, 237, 274, 314, 358, 406, 458, 514, 575, 640, 710, 785, 865, 950, 1041, 1137, 1239, 1347, 1461, 1581, 1708, 1841, 1981, 2128, 2282, 2443, 2612, 2788, 2972, 3164, 3364, 3572, 3789, 4014

Offset: 0

N. J. A. Sloane

Molien series for 4-dimensional representation of S_3 [Nebe, Rains, Sloane, Chap. 7].
From Thomas Wieder, Feb 11 2007: (Start)
If P(i,k) denotes the number of integer partitions of i into k parts and if k=3, then a(n) = Sum_{i=k..n+2} P(i,k). See also A002620 = Quarter-squares, this sequence follows for k=2 as pointed out by Rick L. Shepherd, Feb 27 2004.
For example, a(n=6)=16 because there are 16 integer partitions of n=3,4,...,n+2=8 with k=3 parts:
[[1, 1, 1]],
[[2, 1, 1]],
[[3, 1, 1], [2, 2, 1]]
[[4, 1, 1], [3, 2, 1], [2, 2, 2]],
[[5, 1, 1], [4, 2, 1], [3, 3, 1], [3, 2, 2]],
[[6, 1, 1], [5, 2, 1], [4, 3, 1], [4, 2, 2], [3, 3, 2]]. (End)
Let P(i,k) be the number of integer partitions of n into k parts. Then if k=3 we have a(n) = Sum_{i=k..n} P(i,k=3). - Thomas Wieder, Feb 20 2007
Number of equivalence classes of 3 X n binary matrices when one can permute rows, permute columns and complement columns. - Max Alekseyev, Feb 05 2010
Convolution of the sequences whose n-th terms are given by 1+[n/2] and 1+[n/3], where []=floor. - Clark Kimberling, May 28 2012
Number of partitions of n into two sorts of 1, and one sort each of 2 and 3. - Joerg Arndt, May 05 2014
a(n-3) is the number of partitions mu of 2n of length 4 such that mu has an even number of even entries and the transpose of mu has an even number of even entries (see below example). - John M. Campbell, Feb 03 2016
Number of partitions of 2n+8 into 4 parts such that the sum of the smallest two parts and the sum of the largest two parts are both odd. Also, number of partitions of 2n+4 into 4 parts such that the sum of the smallest two parts and the sum of the largest two parts are both even. - Wesley Ivan Hurt, Jan 19 2021

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 16*x^5 + 23*x^6 + 31*x^7 + ...
From _John M. Campbell_, Feb 03 2016:  (Start)
For example, letting n=6, there are a(n-3)=a(3)=7 partitions mu of 12 of length 4 such mu has an even number of even entries and the transpose of mu has an even number of even entries: (8,2,1,1), (6,4,1,1), (6,3,2,1), (6,2,2,2), (4,4,3,1), (4,4,2,2), (4,3,3,2). For example, the partition
   oooooo
   ooo
   oo
   o
has 2 even entries and the transpose
   oooo
   ooo
   oo
   o
   o
   o
has an even number of even entries. (End)

A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
A. Cayley, Numerical tables supplementary to second memoir on quantics, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, pp. 276-281. [Annotated scanned copy]
L. Colmenarejo, Combinatorics on several families of Kronecker coefficients related to plane partitions, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
Florent de Dinechin, Matei Istoan, Guillaume Sergent, Kinga Illyes, Bogdan Popa and Nicolas Brunie, Arithmetic around the bit heap, HAL Id: ensl-00738412, 2012. - From _N. J. A. Sloane_, Dec 31 2012
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051. doi:10.1109/T-C.1973.223649 - _Max Alekseyev_, Feb 05 2010
H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085.
H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 196
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,-1,0,2,-1)
Index entries for two-way infinite sequences

Cf. A002620, A006148, A006383.

K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; U:=MatrixGroup<4,K|q2,h>; G:=MatrixGroup<4,K|q1,q2,h>; H:=MatrixGroup<4,K|q1,q2,h,p1>; MolienSeries(U);

A000601:=1/(z+1)/(z**2+z+1)/(z-1)**4; # Simon Plouffe in his 1992 dissertation
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+1), right=Set(U, card=1)}, unlabeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=3..52) ; # Zerinvary Lajos, Feb 07 2008

CoefficientList[Series[1/((1-x)^2*(1-x^2)*(1-x^3)), {x, 0, 49}], x] (* Jean-François Alcover, Jul 20 2011 *)
LinearRecurrence[{2,0,-1,-1,0,2,-1},{1,2,4,7,11,16,23},50] (* Harvey P. Dale, Mar 17 2013 *)
a[ n_] := Quotient[ 2 n^3 + 21 n^2 + 66 n, 72] + 1; (* Michael Somos, May 28 2014 *)

Vec(1/((1-x)^2*(1-x^2)*(1-x^3))+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

{a(n) = (2*n^3 + 21*n^2 + 66*n) \ 72 + 1}; /* Michael Somos, May 28 2014 */

a(n) = n^3/36 +7*n^2/24 +11*n/12 +119/144 +(-1)^n/16 + A057078(n)/9. - R. J. Mathar, Mar 14 2011
a(0)=1, a(1)=2, a(2)=4, a(3)=7, a(4)=11, a(5)=16, a(6)=23, a(n) = 2*a(n-1) - a(n-3) - a(n-4) + 2*a(n-6) - a(n-7). - Harvey P. Dale, Mar 17 2013
It appears that a(n) = ((4*n^3+42*n^2+140*n+102+21*(1+(-1)^n))/8-6*floor((2*n+5+3*(-1)^n)/12))/18. - Luce ETIENNE, May 05 2014
Euler transform of length 3 sequence [ 2, 1, 1]. - Michael Somos, May 28 2014
a(-7 - n) = -a(n). - Michael Somos, May 28 2014

More terms from James Sellers, Feb 06 2000

A006380 Number of equivalence classes of 4 X n binary matrices when one can permute rows, permute columns and complement columns.

Original entry on oeis.org

1, 3, 8, 19, 41, 81, 153, 273, 468, 774, 1240, 1930, 2933, 4356, 6341, 9064, 12743, 17643, 24093, 32479, 43270, 57019, 74377, 96103, 123089, 156354, 197081, 246622, 306519, 378520, 464614, 567028, 688276, 831169, 998845, 1194793, 1422899, 1687447, 1993182

Offset: 0

N. J. A. Sloane

M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
Index entries for sequences related to binary matrices
Index entries for linear recurrences with constant coefficients, signature (4,-5,2,-2,2,5,-8,6,-8,5,2,-2,2,-5,4,-1).
Index entries for two-way infinite sequences

Row n=4 of A363349.

Cf. A000601, A006148, A006383.

LinearRecurrence[{4,-5,2,-2,2,5,-8,6,-8,5,2,-2,2,-5,4,-1},{1,3,8,19,41,81,153,273,468,774,1240,1930,2933,4356,6341,9064},40] (* Harvey P. Dale, Nov 23 2024 *)

Vec((1 - x + x^2 + x^4 + x^6 - x^7 + x^8)/((1 - x)^8*(1 + x)^2*(1 + x^2)*(1 + x + x^2)^2) + O(x^41)) \\ Andrew Howroyd, May 30 2023

G.f.: (1 - x + x^2 + x^4 + x^6 - x^7 + x^8)/((1 - x)^8*(1 + x)^2*(1 + x^2)*(1 + x + x^2)^2). - Andrew Howroyd, May 30 2023

Terms a(7) onwards from Max Alekseyev, Feb 05 2010

A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 4, 4, 3, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 6, 11, 19, 10, 4, 1, 1, 1, 7, 16, 41, 32, 16, 4, 1, 1, 1, 8, 23, 81, 101, 68, 20, 5, 1, 1, 1, 9, 31, 153, 299, 301, 114, 29, 5, 1, 1, 1, 10, 41, 273, 849, 1358, 757, 210, 35, 6, 1

Offset: 0

Andrew Howroyd, May 28 2023

T(n,k) is also the number of equivalence classes of n X k binary matrices with an even number of 1's in each column under permutation of rows and columns.

			Array begins:
======================================================
n/k| 0 1  2   3    4     5      6       7        8 ...
---+--------------------------------------------------
0  | 1 1  1   1    1     1      1       1        1 ...
1  | 1 1  1   1    1     1      1       1        1 ...
2  | 1 2  3   4    5     6      7       8        9 ...
3  | 1 2  4   7   11    16     23      31       41 ...
4  | 1 3  8  19   41    81    153     273      468 ...
5  | 1 3 10  32  101   299    849    2290     5901 ...
6  | 1 4 16  68  301  1358   6128   27008   114763 ...
7  | 1 4 20 114  757  5567  43534  343656  2645494 ...
8  | 1 5 29 210 1981 23350 319119 4633380 67013431 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)

A259344 is the same array without the first row and column read by upward antidiagonals.
Columns k=0..6 are A000012, A004526(n+2), A005232, A006381, A006382, A056204, A056205.
Rows n=2..4 are A000027(n+1), A000601, A006380.
Main diagonal is A006383.
Cf. A028657, A241956, A362905.

\\ Compare A028657.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={sum(j=1, #q, gcd(t, q[j]))}
T(n, k)={if(n==0, 1, my(s=0); forpart(q=n, my(e=1<

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A000601 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^3)).

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A006380 Number of equivalence classes of 4 X n binary matrices when one can permute rows, permute columns and complement columns.

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A363349 Array read by antidiagonals: T(n,k) is the number of equivalence classes of n X k binary matrices under permutation of rows and columns and complementation of columns.

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A363351 Number of n element multisets of length n vectors over GF(2) that sum to zero.

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A259344 Array read by antidiagonals: number of inequivalent m X n (0,1)-matrices under permutation of rows and permutation and/or complementation of columns.

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A000601 Expansion of 1/((1-x)^2(1-x^2)(1-x^3)).