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%I M3460 N1407 #98 Feb 28 2023 21:09:45
%S 1,4,13,36,87,190,386,734,1324,2284,3790,6080,9473,14378,21323,30974,
%T 44159,61898,85440,116286,156240,207446,272432,354162,456097,582238,
%U 737205,926298,1155567,1431892,1763074,2157904,2626276,3179278,3829294,4590118,5477081
%N Number of 3 X n binary matrices up to row and column permutations.
%C Also, number of unlabeled bipartite graphs with three left vertices and n right vertices. - _Yavuz Oruc_, Jan 22 2018
%D A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002727/b002727.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Atmaca, and A. Yavuz Oruç, <a href="https://doi.org/10.1016/j.akcej.2017.11.008">On the size of two families of unlabeled bipartite graphs, AKCE International Journal of Graphs and Combinatorics, 2017.</a>
%H M. A. Harrison, <a href="http://dx.doi.org/10.1109/T-C.1973.223649">On the number of classes of binary matrices</a>, IEEE Trans. Computers, 22 (1973), 1048-1051.
%H M. A. Harrison, <a href="/A000711/a000711.pdf">On the number of classes of binary matrices</a>, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
%H Vladeta Jovovic, <a href="/A005748/a005748.pdf">Binary matrices up to row and column permutations</a>
%H A. Kerber, <a href="/A002727/a002727.pdf">Experimentelle Mathematik</a>, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
%H B. Misek, <a href="http://dml.cz/dmlcz/108444">On the number of classes of strongly equivalent incidence matrices</a>, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (4, -4, -2, 2, 4, 3, -12, 3, 4, 2, -2, -4, 4, -1).
%F 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.
%F 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
%F a(n) = -a(-8 - n) for all n in Z. - _Michael Somos_, Aug 22 2016
%F From _Yavuz Oruc_, Jan 22 2018: (Start)
%F 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).
%F 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).
%F 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)
%e G.f. = 1 + 4*x + 13*x^2 + 36*x^3 + 87*x^4 + 190*x^5 + 386*x^6 + 734*x^7 + ...
%t 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 *)
%t Table[Which[
%t Mod[n, 3] == 0,
%t 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]),
%t Mod[n, 3] == 1,
%t 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]),
%t Mod[n, 3] == 2,
%t 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])
%t ], {n, 0, 100}] (* _Yavuz Oruc_, Jan 22 2018 *)
%o (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
%o (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 */
%o (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
%o (PARI) Vec(G(3, x) + O(x^40)) \\ G defined in A028657. - _Andrew Howroyd_, Feb 28 2023
%Y Cf. A000601, A002623, A006148, A006381.
%Y A row of the array A(m,n) described in A028657. - _N. J. A. Sloane_, Sep 01 2013
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Feb 04 2000
%E Definition corrected by _Max Alekseyev_, Feb 05 2010
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