A053497
Number of degree-n permutations of order dividing 7.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 721, 5761, 25921, 86401, 237601, 570241, 1235521, 892045441, 13348249201, 106757164801, 604924594561, 2722120577281, 10344007402561, 34479959558401, 24928970490633601, 546446134633639681, 6281586217487489041, 50248618811434961281
Offset: 0
- R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.
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R:=PowerSeriesRing(Rationals(), 31); Coefficients(R!(Laplace( Exp(x + x^7/7) ))); // G. C. Greubel, May 14 2019, Mar 07 2021
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a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 7])))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Feb 14 2013
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CoefficientList[Series[Exp[x+x^7/7], {x, 0, 24}], x]*Range[0, 24]! (* Jean-François Alcover, Mar 24 2014 *)
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my(x='x+O('x^30)); Vec(serlaplace( exp(x+x^7/7) )) \\ G. C. Greubel, May 14 2019
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f=factorial; [sum(f(n)/(7^j*f(j)*f(n-7*j)) for j in (0..n/7)) for n in (0..30)] # G. C. Greubel, May 14 2019
A118933
Triangle, read by rows, where T(n,k) = n!/(k!*(n-4*k)!*4^k) for n>=4*k>=0.
Original entry on oeis.org
1, 1, 1, 1, 1, 6, 1, 30, 1, 90, 1, 210, 1, 420, 1260, 1, 756, 11340, 1, 1260, 56700, 1, 1980, 207900, 1, 2970, 623700, 1247400, 1, 4290, 1621620, 16216200, 1, 6006, 3783780, 113513400, 1, 8190, 8108100, 567567000, 1, 10920, 16216200, 2270268000, 3405402000
Offset: 0
Triangle begins:
1;
1;
1;
1;
1, 6;
1, 30;
1, 90;
1, 210;
1, 420, 1260;
1, 756, 11340;
1, 1260, 56700;
1, 1980, 207900;
1, 2970, 623700, 1247400; ...
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F:= Factorial;
[n lt 4*k select 0 else F(n)/(4^k*F(k)*F(n-4*k)): k in [0..Floor(n/4)], n in [0..20]]; // G. C. Greubel, Mar 07 2021
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T[n_, k_]:= If[n<4*k, 0, n!/(4^k*k!*(n-4*k)!)];
Table[T[n, k], {n,0,20}, {k,0,n/4}]//Flatten (* G. C. Greubel, Mar 07 2021 *)
-
T(n,k)=if(n<4*k,0,n!/(k!*(n-4*k)!*4^k))
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f=factorial;
flatten([[0 if n<4*k else f(n)/(4^k*f(k)*f(n-4*k)) for k in [0..n/4]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021
A293588
E.g.f.: exp(x + x^6/6).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 121, 841, 3361, 10081, 25201, 55441, 6763681, 86692321, 605765161, 3027624601, 12109056961, 41169011521, 5063607974881, 94197184734241, 939457659787201, 6572292677455681, 36141156689382361, 166238526616664041, 20612479896229156321
Offset: 0
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F:= Factorial;
[(&+[ F(n)/(6^k*F(k)*F(n-6*k)): k in [0..Floor(n/3)]]): n in [0..30]]; // G. C. Greubel, Mar 07 2021
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With[{nn=30},CoefficientList[Series[Exp[x+x^6/6],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Dec 11 2017 *)
Table[Sum[n!/(6^k*k!*(n-6*k)!), {k, 0, n/6}], {n, 0, 30}] (* G. C. Greubel, Mar 07 2021 *)
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my(x = 'x + O('x^30)); Vec(serlaplace(exp(x + x^6/6))) \\ Michel Marcus, Oct 13 2017
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f=factorial;
[sum( f(n)/(6^k*f(k)*f(n-6*k)) for k in [0..n/3]) for n in [0..30]] # G. C. Greubel, Mar 07 2021
Showing 1-3 of 3 results.
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