A138772 Number of entries in the second cycles of all permutations of {1,2,...,n}; each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.
0, 1, 5, 27, 168, 1200, 9720, 88200, 887040, 9797760, 117936000, 1536796800, 21555072000, 323805081600, 5187108326400, 88268019840000, 1590132031488000, 30233431388160000, 605024315191296000, 12711912992722944000, 279783730940313600000, 6437458713635389440000
Offset: 1
Keywords
Examples
a(3) = 5 because the number of entries in the second cycles of (1)(2)(3), (1)(23), (132), (12)(3), (123) and (13)(2) is 1+2+0+1+0+1=5.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..200
Programs
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GAP
List([1..30], n-> (n-1)*(n+2)*Factorial(n-1)/4); # G. C. Greubel, Jul 07 2019
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Magma
[(n-1)*(n+2)*Factorial(n-1)/4: n in [1..30]]; // G. C. Greubel, Jul 07 2019
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Maple
seq((1/4)*factorial(n-1)*(n-1)*(n+2), n = 1 .. 30);
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Mathematica
Table[((1/4)(n-1)!(n-1)(n+2)),{n,1,30}] (* Vincenzo Librandi, May 14 2012 *) -
PARI
vector(30, n, (n-1)*(n+2)*(n-1)!/4) \\ G. C. Greubel, Jul 07 2019
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Sage
[(n-1)*(n+2)*factorial(n-1)/4 for n in (1..30)] # G. C. Greubel, Jul 07 2019
Formula
a(n) = (1/4)*(n-1)!*(n-1)*(n+2).
a(n) = (n+1)*a(n-1) + (n-2)!.
a(n) = (n-1)*a(n-1) + n!/2.
a(n) = Sum_{k=0..n-1} k*A138771(n,k).
E.g.f. if offset 0: x*(2-x)/(2*(1-x)^3). Such e.g.f. computations resulted from e-mail exchange with Gary Detlefs. - Wolfdieter Lang, May 27 2010
a(n) = n! * Sum_{i=1..n} (Sum_{j=1..i} (j/i)). - Pedro Caceres, Apr 19 2019
E.g.f.: ( x*(2-x)/(1-x)^2 + 2*log(1-x) )/4. - G. C. Greubel, Jul 07 2019
D-finite with recurrence a(n) +(-n-1)*a(n-1) -2*a(n-2) +2*(n-3)*a(n-3)=0. - R. J. Mathar, Jul 26 2022