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A002727 Number of 3 X n binary matrices up to row and column permutations.
(Formerly M3460 N1407)
24
1, 4, 13, 36, 87, 190, 386, 734, 1324, 2284, 3790, 6080, 9473, 14378, 21323, 30974, 44159, 61898, 85440, 116286, 156240, 207446, 272432, 354162, 456097, 582238, 737205, 926298, 1155567, 1431892, 1763074, 2157904, 2626276, 3179278, 3829294, 4590118, 5477081 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Also, number of unlabeled bipartite graphs with three left vertices and n right vertices. - Yavuz Oruc, Jan 22 2018
REFERENCES
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83.
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).
LINKS
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)
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Index entries for linear recurrences with constant coefficients, signature (4, -4, -2, 2, 4, 3, -12, 3, 4, 2, -2, -4, 4, -1).
FORMULA
G.f.: (x^6+x^4+2*x^3+x^2+1)/((1-x)^4*(1-x^2)^2*(1-x^3)^2). - Vladeta Jovovic, Feb 04 2000.
a(0)=1, a(1)=4, a(2)=13, a(3)=36, a(4)=87, a(5)=190, a(6)=386, a(7)=734, a(8)=1324, a(9)=2284, a(10)=3790, a(11)=6080, a(12)=9473, a(13)=14378. For n>13, a(n)=4*a(n-1)-4*a(n-2)-2*a(n-3)+2*a(n-4)+4*a(n-5)+3*a(n-6)- 12*a(n-7)+ 3*a(n-8)+4*a(n-9)+2*a(n-10)-2*a(n-11)-4*a(n-12)+4*a(n-13)-a(n-14). - Harvey P. Dale, Nov 10 2011
a(n) = -a(-8 - n) for all n in Z. - Michael Somos, Aug 22 2016
From Yavuz Oruc, Jan 22 2018: (Start)
If n == 0 (mod 3) then a(n)=(1/6)*(binomial(n+7,7) + (3(n+4)(2n^4 + 32n^3 + 172n^2 + 352n + 15(-1)^n + 225))/960 + (2(n^3 + 12n^2 + 45n + 54))/54).
If n == 1 (mod 3) then a(n)=(1/6)*(binomial(n+7,7) + (3(n+4)(2n^4 + 32n^3 + 172n^2 + 352n + 15(-1)^n + 225))/960 + (2(n^3 + 12n^2 + 45n + 50))/54).
If n == 2 (mod 3) then a(n)=(1/6)*(binomial(n+7,7) + (3(n+4)(2n^4 + 32n^3 + 172n^2 + 352n + 15(-1)^n + 225))/960 + (2(n^3 + 12n^2 + 39n + 28))/54). (End)
EXAMPLE
G.f. = 1 + 4*x + 13*x^2 + 36*x^3 + 87*x^4 + 190*x^5 + 386*x^6 + 734*x^7 + ...
MATHEMATICA
CoefficientList[Series[(x^6+x^4+2x^3+x^2+1)/((1-x)^4(1-x^2)^2(1-x^3)^2), {x, 0, 40}], x] (* or *) LinearRecurrence[{4, -4, -2, 2, 4, 3, -12, 3, 4, 2, -2, -4, 4, -1}, {1, 4, 13, 36, 87, 190, 386, 734, 1324, 2284, 3790, 6080, 9473, 14378}, 41] (* Harvey P. Dale, Nov 10 2011 *)
Table[Which[
Mod[n, 3] == 0,
1/6 (1/27 (54 + 45 n + 12 n^2 + n^3) + 1/320 (4 + n) *(225 + 15 (-1)^n + 352 n + 172 n^2 + 32 n^3 + 2 n^4) + Binomial[7 + n, 7]),
Mod[n, 3] == 1,
1/6 (1/27 (50 + 45 n + 12 n^2 + n^3) + 1/320 (4 + n) *(225 + 15 (-1)^n + 352 n + 172 n^2 + 32 n^3 + 2 n^4) + Binomial[7 + n, 7]),
Mod[n, 3] == 2,
1/6 (1/27 (28 + 39 n + 12 n^2 + n^3) + 1/320 (4 + n) *(225 + 15 (-1)^n + 352 n + 172 n^2 + 32 n^3 + 2 n^4) + Binomial[7 + n, 7])
], {n, 0, 100}] (* Yavuz Oruc, Jan 22 2018 *)
PROG
(Magma) I:=[1, 4, 13, 36, 87, 190, 386, 734, 1324, 2284, 3790, 6080, 9473, 14378]; [n le 14 select I[n] else 4*Self(n-1)-4*Self(n-2)-2*Self(n-3)+2*Self(n-4)+4*Self(n-5)+3*Self(n-6)-12*Self(n-7)+ 3*Self(n-8)+4*Self(n-9)+2*Self(n-10)-2*Self(n-11)-4*Self(n-12)+4*Self(n-13)-Self(n-14): n in [1..50]]; // Vincenzo Librandi, Oct 13 2015
(PARI) {a(n) = (6*n^7 + 168*n^6 + 2121*n^5 + 15540*n^4 + 70084*n^3 + 190512*n^2 + n*[284544, 281709, 277824, 281709, 284544, 274989][n%6+1]) \ 181440 + 1}; /* Michael Somos, Aug 22 2016 */
(PARI) x='x+O('x^99); Vec((1+x^2+2*x^3+x^4+x^6)/((1-x)^2*((1-x)*(1-x^2)*(1-x^3))^2)) \\ Altug Alkan, Mar 03 2018
(PARI) Vec(G(3, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023
CROSSREFS
A row of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013
Sequence in context: A270988 A272556 A173723 * A320589 A328542 A036629
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Feb 04 2000
Definition corrected by Max Alekseyev, Feb 05 2010
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)