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 A060244 Triangle a(n,k) of bipartite partitions of n objects into parts >1, k of which are black. 8

%I

%S 1,0,0,1,1,1,1,1,1,1,2,2,3,2,2,2,3,4,4,3,2,4,5,8,8,8,5,4,4,7,11,13,13,

%T 11,7,4,7,11,19,22,26,22,19,11,7,8,15,26,35,40,40,35,26,15,8,12,22,41,

%U 54,69,70,69,54,41,22,12,14,30,56,81,104,116,116,104,81,56,30,14,21,42

%N Triangle a(n,k) of bipartite partitions of n objects into parts >1, k of which are black.

%D P. A. MacMahon, Memoir on symmetric functions of the roots of systems of equations, Phil. Trans. Royal Soc. London, 181 (1890), 481-536; Coll. Papers II, 32-87.

%F G.f.: Product_{ i=2..infinity, j=0..i} 1/(1-x^(i-j)*y^j).

%e Series ends ... + 2*x^5 + 3*x^4*y + 4*x^3*y^2 + 4*x^2*y^3 + 3*x*y^4 + 2*y^5 + 2*x^4 + 2*x^3*y + 3*x^2*y^2 + 2*x*y^3 + 2*y^4 + x^3 + x^2*y + x*y^2 + y^3 + x^2 + x*y + y^2 + 1.

%e 1;

%e 0, 0;

%e 1, 1, 1;

%e 1, 1, 1, 1;

%e 2, 2, 3, 2, 2;

%e ...

%p read transforms; t1 := mul( mul( 1/(1-x^(i-j)*y^j), j=0..i), i=2..11): SERIES2(t1,x,y,7);

%t max = 12; gf = Product[1/(1 - x^(i - j)*y^j), {i, 2, max}, {j, 0, i}]; se = Series[gf, {x, 0, max}, {y, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[se, {x, 0, n}, {y, 0, k}]; Flatten[ Table[ t[n - k, k], {n, 0, max}, {k, 0, n}]] (* _Jean-François Alcover_, after Maple *)

%Y Columns 0-6: A002865, A000041, A024786, A291553, A291589, A291590, A291596.

%Y Row sums: A060285.

%Y Cf. A005380, A054225.

%K nonn,nice,tabl,easy

%O 0,11

%A _N. J. A. Sloane_, Mar 22 2001

%E More terms from _Vladeta Jovovic_, Mar 23 2001

%E Edited by _Christian G. Bower_, Jan 08 2004

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Last modified May 20 21:27 EDT 2019. Contains 323426 sequences. (Running on oeis4.)