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

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

Offset: 0

Author

N. J. A. Sloane, Mar 22 2001

Keywords

• nonn
• nice
• tabl
• easy

Examples

			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.
1;
0, 0;
1, 1, 1;
1, 1, 1, 1;
2, 2, 3, 2, 2;
...
		

References

• 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.

Crossrefs

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

Row sums: A060285.

Cf. A005380, A054225.

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

More terms from Vladeta Jovovic, Mar 23 2001

Edited by Christian G. Bower, Jan 08 2004