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A215718 Number of permutations on n points admitting a seventh root. 7
1, 1, 2, 6, 24, 120, 720, 4320, 34560, 311040, 3110400, 34214400, 410572800, 5337446400, 69386803200, 1040802048000, 16652832768000, 283098157056000, 5095766827008000, 96819569713152000, 1936391394263040000, 38727827885260800000, 852012213475737600000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the number of permutations of n points such that for all positive m, the number of (7m)-cycles is a multiple of 7.

LINKS

Eric M. Schmidt, Table of n, a(n) for n = 0..200

H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 148-149, Thms. 4.8.2 and 4.8.3.

FORMULA

E.g.f.: (1 - x^7)^(1/7)/(1 - x) * Product_{m>=1} E_7(x^(7m)/(7m)) ), where E_7(x) = 1 + x^7/7! + x^14/14! + ... .

MAPLE

with(combinat):

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(`if`(irem(j, igcd(i, 7))<>0, 0, (i-1)!^j*

      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))

    end:

a:= n-> b(n$2):

seq(a(n), n=0..25);  # Alois P. Heinz, Sep 08 2014

MATHEMATICA

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[j, GCD[i, 7]] != 0, 0, (i-1)!^j*multinomial[n, Prepend[Table[i, {j}], n-i*j]]/j!*b[n-i*j, i - 1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 21 2016, after Alois P. Heinz *)

PROG

(PARI)

{ A215718_list(numterms) = Vec(serlaplace((1 - x^7 + O(x^numterms))^(1/7)/(1-x) * prod(m=1, numterms\7, exp7(x^(7*m)/(7*m), numterms\(7*m)+1)))); }

{ exp7(y, prec) = subst(serconvol(exp(x + O(x^prec)), 1/(1-x^7) + O(x^prec)), x, y); }

CROSSREFS

Cf. A003483, A103619, A103620, A215716, A215717.

Column k=7 of A247005.

Sequence in context: A179365 A070947 A267386 * A060727 A152350 A152368

Adjacent sequences:  A215715 A215716 A215717 * A215719 A215720 A215721

KEYWORD

nonn

AUTHOR

Eric M. Schmidt, Aug 23 2012

STATUS

approved

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Last modified April 9 02:12 EDT 2020. Contains 333339 sequences. (Running on oeis4.)