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Search: id:A060243
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| A060243 |
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Triangle a(n,k) of bipartite partitions of n objects, k of which are black. |
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+0
23
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| 1, 1, 1, 2, 2, 2, 3, 4, 4, 3, 5, 7, 9, 7, 5, 7, 12, 16, 16, 12, 7, 11, 19, 29, 31, 29, 19, 11, 15, 30, 47, 57, 57, 47, 30, 15, 22, 45, 77, 97, 109, 97, 77, 45, 22, 30, 67, 118, 162, 189, 189, 162, 118, 67, 30, 42, 97, 181, 257, 323, 339, 323, 257, 181, 97, 42, 56, 139, 267
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Number of ways to factor p^(n-k)*q^k where p and q are distinct primes.
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REFERENCES
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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.
M. S. Cheema, Tables of partitions of Gaussian integers, National Institute of Sciences of India, New Delhi, 1956.
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FORMULA
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G.f.: Product_{ i=1..infinity, j=0..i} 1/(1-x^(i-j)*y^j).
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EXAMPLE
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Series ends ... + 7*x^5 + 12*x^4*y + 16*x^3*y^2 + 16*x^2*y^3 + 12*x*y^4 + 7*y^5 + 5*x^4 + 7*x^3*y + 9*x^2*y^2 + 7*x*y^3 + 5*y^4 + 3*x^3 + 4*x^2*y + 4*x*y^2 + 3*y^3 + 2*x^2 + 2*x*y + 2*y^2 + x + y + 1.
1; 1,1; 2,2,2; 3,4,4,3; ...
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MAPLE
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read transforms; t1 := mul( mul( 1/(1-x^(i-j)*y^j), j=0..i), i=1..11): SERIES2(t1, x, y, 6);
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PROGRAM
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(PARI) {T(n, k)=if(n<0|k<0, 0, polcoeff( polcoeff( prod(i=1, n, prod(j=0, i, 1/(1-x^i*y^j), 1+O(x^n)*x)), n), k))} /* Michael Somos Apr 19 2005 */
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CROSSREFS
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Columns 0-10: A000041, A000070, A000291, A000412, A000465, A000491, A002755-A002759.
Row sums: A005380. a(2n, n): A002774. a(n, [n/2]): A091437. Cf. A060244.
Sequence in context: A051601 A054225 A074829 this_sequence A091822 A060973 A097915
Adjacent sequences: A060240 A060241 A060242 this_sequence A060244 A060245 A060246
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KEYWORD
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nonn,nice,tabl,easy
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AUTHOR
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njas, Mar 22 2001
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 23 2001
Edited by Christian G. Bower (bowerc(AT)usa.net), Jan 08 2004
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