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%I M2452 N0973 #32 Jun 25 2023 02:56:07
%S 1,1,3,5,9,13,22,30,45,61,85,111,150,190,247,309,390,478,593,715,870,
%T 1038,1243,1465,1735,2023,2368,2740,3175,3643,4189,4771,5443,6163,
%U 6982,7858,8852,9908,11098,12366,13780,15284,16958,18730,20692,22772,25058,27478
%N Number of two-rowed partitions of length 3.
%D G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
%D A. Cayley, Calculation of the minimum N.G.F. of the binary seventhic, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419.
%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 Alois P. Heinz, <a href="/A001993/b001993.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Cayley, <a href="/A001993/a001993.pdf">Calculation of the minimum N.G.F. of the binary seventhic</a>, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 10, p. 408-419. [Annotated scanned copy]
%H L. Colmenarejo, <a href="http://arxiv.org/abs/1604.00803">Combinatorics on several families of Kronecker coefficients related to plane partitions</a>, arXiv:1604.00803 [math.CO], 2016. See Table 1 p. 5.
%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (1, 2, 0, -2, -4, 1, 3, 3, 1, -4, -2, 0, 2, 1, -1).
%F G.f.: 1/((1-x)*(1-x^2)^2*(1-x^3)^2*(1-x^4)).
%p a:= n-> (Matrix(15, (i,j)-> if (i=j-1) then 1 elif j=1 then [1, 2, 0, -2, -4, 1, 3, 3, 1, -4, -2, 0, 2, 1, -1][i] else 0 fi)^n)[1,1]: seq(a(n), n=0..50); # _Alois P. Heinz_, Jul 31 2008
%t a[n_] := (Table[Which[i == j-1, 1, j == 1, {1, 2, 0, -2, -4, 1, 3, 3, 1, -4, -2, 0, 2, 1, -1}[[i]], True, 0], {i, 1, 15}, {j, 1, 15}] // MatrixPower[#, n]&)[[1, 1]]; Table[a[n], {n, 0, 46}] (* _Jean-François Alcover_, Mar 17 2014, after _Alois P. Heinz_ *)
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _James A. Sellers_, Feb 09 2000
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