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

0,11

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.

Links

Table of n, a(n) for n=0..79.

Formula

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

Example

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;
...

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 *)

Crossrefs

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

Row sums: A060285.

Cf. A005380, A054225.

Sequence in context: A085962 A160821 A300225 * A072814 A196229 A191302

Adjacent sequences: A060241 A060242 A060243 * A060245 A060246 A060247

Keyword

nonn,nice,tabl,easy

Author

N. J. A. Sloane, Mar 22 2001

Extensions

More terms from Vladeta Jovovic, Mar 23 2001

Edited by Christian G. Bower, Jan 08 2004

Status

approved