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A007870 Determinant of character table of symmetric group S_n. 5
1, 1, 2, 6, 96, 2880, 9953280, 100329062400, 10651768002183168000, 150283391703941024789299200000, 9263795272057860957392207640004657152000000000, 16027108137650009941734148595388542471170145479274004480000000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..18

F. W. Schmidt and R. Simion, On a partition identity, J. Combin. Theory, A 36 (1984), 249-252.

D. Vaintrob, A product identity for partitions, MathOverflow, June 2012.

FORMULA

Product of all parts of all partitions of n.

EXAMPLE

1 + x + 2*x^2 + 6*x^3 + 96*x^4 + 2880*x^5 + 9953280*x^6 + 100329062400*x^7 + ...

MAPLE

b:= proc(n, i) option remember; `if`(n=0, [1$2], ((f, g)->

       [f[1]+g[1], f[2]*g[2]*i^g[1]])(`if`(i<2, [0, 1],

       b(n, i-1)), `if`(i>n, [0, 1], b(n-i, i))))

    end:

a:= n-> b(n, n)[2]:

seq(a(n), n=0..12);  # Alois P. Heinz, Jul 30 2013

# Alternative:

with(combinat): P:=proc(n) local a, k; a:=partition(n);

a:=mul(convert(a[k], `*`), k=1..nops(a)); end: seq(P(i), i=0..12);

# Paolo P. Lava, Dec 21 2018

MATHEMATICA

Needs["Combinatorica`"]; Table[Times@@Flatten[Partitions[n]], {n, 10}]

a[ n_] := If[n < 0, 0, Times @@ Flatten @ IntegerPartitions @ n] (* Michael Somos, Jun 11 2012 *)

Table[Exp[Total[Map[Log, IntegerPartitions [n]], 2]], {n, 1, 25}] (* Richard R. Forberg, Dec 08 2014 *)

b[n_, i_] := b[n, i] = If[n == 0, {1, 1}, Function[{f, g}, {f[[1]] + g[[1]], f[[2]]*g[[2]]*i^g[[1]]}][If[i < 2, {0, 1}, b[n, i - 1]], If[i > n, {0, 1}, b[n - i, i]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

PROG

(GAP) List(List([0..11], n->Flat(Partitions(n))), Product); # Muniru A Asiru, Dec 21 2018

CROSSREFS

Cf. A086644, A006906.

Sequence in context: A092287 A035482 A322716 * A081992 A066091 A100704

Adjacent sequences:  A007867 A007868 A007869 * A007871 A007872 A007873

KEYWORD

nonn

AUTHOR

Peter J. Cameron, Götz Pfeiffer [ goetz(AT)dcs.st-and.ac.uk ]

STATUS

approved

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Last modified March 24 00:12 EDT 2019. Contains 321444 sequences. (Running on oeis4.)