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A102189
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Array of multinomial numbers (row reversed order of table A036039).
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10
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1, 1, 1, 1, 3, 2, 1, 6, 3, 8, 6, 1, 10, 15, 20, 20, 30, 24, 1, 15, 45, 40, 15, 120, 90, 40, 90, 144, 120, 1, 21, 105, 70, 105, 420, 210, 210, 280, 630, 504, 420, 504, 840, 720, 1, 28, 210, 112, 420, 1120, 420, 105, 1680, 1120, 2520, 1344, 1120, 1260, 3360, 4032, 3360
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OFFSET
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1,5
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COMMENTS
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See Abramowitz and Stegun, Handbook, p. 831, column labeled "M_2", read backwards.
The sequence of row lengths is [1,2,3,5,7,11,15,...] = A000041(n), n>=1 (partition numbers).
Row n of this array gives the coefficients of the cycle index polynomial n!*Z(S_n) for the symmetric group S_n. For instance, Z(S_4)= (x[1]^4 + 6*x[1]^2*x[2] + 3*x[2]^2 + 8*x[1]*x[3] + 6*x[4])/4!. The partitions of 4 appear here in the reversed Abramowitz-Stegun order.
See the W. Lang link "Solution of Newton's Identities" and the Note added Jun 06 2007 in the link "More rows and S_n cycle index polynomials" for the appearance of the signed array. - Wolfdieter Lang, Aug 01 2013
Multiplying the values of row n by the corresponding values in row n of A110141, one obtains n!. - Jaimal Ichharam, Aug 06 2015
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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EXAMPLE
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Triangle begins:
[1];
[1,1];
[1,3,2];
[1,6,3,8,6];
[1,10,15,20,20,30,24];
...
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MATHEMATICA
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aspartitions[n_] := Reverse /@ Sort[Sort /@ IntegerPartitions[n]]; ascycleclasses[n_Integer] := n!/(Times @@ #)& /@ ((#! Range[n]^#)& /@ Function[par, Count[par, #]& /@ Range[n]] /@ aspartitions[n]); row[n_] := ascycleclasses[n] // Reverse; Table[row[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Feb 04 2014, after A036039 and Wouter Meeussen *)
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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