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A215718 Number of permutations on n points admitting a seventh root.

%I A215718 #20 Feb 19 2025 05:44:21
%S A215718 1,1,2,6,24,120,720,4320,34560,311040,3110400,34214400,410572800,
%T A215718 5337446400,69386803200,1040802048000,16652832768000,283098157056000,
%U A215718 5095766827008000,96819569713152000,1936391394263040000,38727827885260800000,852012213475737600000
%N A215718 Number of permutations on n points admitting a seventh root.
%C A215718 a(n) is the number of permutations of n points such that for all positive m, the number of (7*m)-cycles is a multiple of 7.
%H A215718 Eric M. Schmidt, <a href="/A215718/b215718.txt">Table of n, a(n) for n = 0..200</a>
%H A215718 H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, p. 148-149, Thms. 4.8.2 and 4.8.3.
%F A215718 E.g.f.: ((1 - x^7)^(1/7)/(1 - x)) * Product_{m>=1} E_7(x^(7*m)/(7*m)), where E_7(x) = 1 + x^7/7! + x^14/14! + ... .
%p A215718 with(combinat):
%p A215718 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A215718       add(`if`(irem(j, igcd(i, 7))<>0, 0, (i-1)!^j*
%p A215718       multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
%p A215718     end:
%p A215718 a:= n-> b(n$2):
%p A215718 seq(a(n), n=0..25);  # _Alois P. Heinz_, Sep 08 2014
%t A215718 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_ *)
%o A215718 (PARI)
%o A215718 { 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)))); }
%o A215718 { exp7(y, prec) = subst(serconvol(exp(x + O(x^prec)), 1/(1-x^7) + O(x^prec)), x, y); }
%Y A215718 Cf. A003483, A103619, A103620, A215716, A215717.
%Y A215718 Column k=7 of A247005.
%K A215718 nonn
%O A215718 0,3
%A A215718 _Eric M. Schmidt_, Aug 23 2012