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A060244
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Triangle a(n,k) of bipartite partitions of n objects into parts >1, k of which are black.
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4
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,11
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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.
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FORMULA
| G.f.: Product_{ i=2..infinity, j=0..i} 1/(1-x^(i-j)*y^j).
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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; ...
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MAPLE
| read transforms; t1 := mul( mul( 1/(1-x^(i-j)*y^j), j=0..i), i=2..11): SERIES2(t1, x, y, 7);
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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}]] (* From Jean-François Alcover, after Maple *)
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CROSSREFS
| Columns 0-2: A002865, A000041, A024786. Row sums: A060285. Cf. A005380, A054225.
Sequence in context: A100549 A085962 A160821 * A196229 A191302 A161189
Adjacent sequences: A060241 A060242 A060243 * A060245 A060246 A060247
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KEYWORD
| nonn,nice,tabl,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mar 22 2001
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 23 2001
Edited by Christian G. Bower (bowerc(AT)usa.net), Jan 08 2004
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