cp's OEIS Frontend

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

%I A060244 #23 Apr 06 2018 18:13:05
%S A060244 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 A060244 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 A060244 54,69,70,69,54,41,22,12,14,30,56,81,104,116,116,104,81,56,30,14,21,42
%N A060244 Triangle a(n,k) of bipartite partitions of n objects into parts >1, k of which are black.
%D A060244 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 A060244 G.f.: Product_{ i=2..infinity, j=0..i} 1/(1-x^(i-j)*y^j).
%e A060244 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 A060244 1;
%e A060244 0, 0;
%e A060244 1, 1, 1;
%e A060244 1, 1, 1, 1;
%e A060244 2, 2, 3, 2, 2;
%e A060244 ...
%p A060244 read transforms; t1 := mul( mul( 1/(1-x^(i-j)*y^j), j=0..i), i=2..11): SERIES2(t1,x,y,7);
%t A060244 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 A060244 Columns 0-6: A002865, A000041, A024786, A291553, A291589, A291590, A291596.
%Y A060244 Row sums: A060285.
%Y A060244 Cf. A005380, A054225.
%K A060244 nonn,nice,tabl,easy
%O A060244 0,11
%A A060244 _N. J. A. Sloane_, Mar 22 2001
%E A060244 More terms from _Vladeta Jovovic_, Mar 23 2001
%E A060244 Edited by _Christian G. Bower_, Jan 08 2004