

A102189


Array of multinomial numbers (row reversed order of table A036039).


10



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

1,5


COMMENTS

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 AbramowitzStegun 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


LINKS

Table of n, a(n) for n=1..61.
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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 8312.
Wolfdieter Lang, More rows and S_n cycle index polynomials.
Wolfdieter Lang, Solution of Newton's Identities.
Andrei Vieru, Analytic renormalization of multiple zeta functions. Geometry and combinatorics of generalized Euler reflection formula for MZV, arXiv preprint arXiv:1601.04703 [math.NT], 2016.


EXAMPLE

Triangle begins:
[1];
[1,1];
[1,3,2];
[1,6,3,8,6];
[1,10,15,20,20,30,24];
...


MATHEMATICA

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 (* JeanFrançois Alcover, Feb 04 2014, after A036039 and Wouter Meeussen *)


CROSSREFS

Cf. A000041, A036039, A110141.
Sequence in context: A324644 A134199 A323417 * A031252 A208152 A194761
Adjacent sequences: A102186 A102187 A102188 * A102190 A102191 A102192


KEYWORD

nonn,easy,tabf


AUTHOR

Wolfdieter Lang, Feb 15 2005


STATUS

approved



