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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.

%I A138772 #48 Nov 22 2024 15:39:46
%S A138772 0,1,5,27,168,1200,9720,88200,887040,9797760,117936000,1536796800,
%T A138772 21555072000,323805081600,5187108326400,88268019840000,
%U A138772 1590132031488000,30233431388160000,605024315191296000,12711912992722944000,279783730940313600000,6437458713635389440000
%N 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.
%H A138772 Vincenzo Librandi, <a href="/A138772/b138772.txt">Table of n, a(n) for n = 1..200</a>
%F A138772 a(n) = (1/4)*(n-1)!*(n-1)*(n+2).
%F A138772 a(n) = (n+1)*a(n-1) + (n-2)!.
%F A138772 a(n) = (n-1)*a(n-1) + n!/2.
%F A138772 a(n) = Sum_{k=0..n-1} k*A138771(n,k).
%F A138772 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
%F A138772 a(n) = A000254(n-1) + A159324(n-1). - _Gary Detlefs_, May 13 2012
%F A138772 a(n) = n! * Sum_{i=1..n} (Sum_{j=1..i} (j/i)). - _Pedro Caceres_, Apr 19 2019
%F A138772 E.g.f.: ( x*(2-x)/(1-x)^2 + 2*log(1-x) )/4. - _G. C. Greubel_, Jul 07 2019
%F A138772 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
%e A138772 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.
%p A138772 seq((1/4)*factorial(n-1)*(n-1)*(n+2), n = 1 .. 30);
%t A138772 Table[((1/4)(n-1)!(n-1)(n+2)),{n,1,30}] (* _Vincenzo Librandi_, May 14 2012 *)
%o A138772 (PARI) vector(30, n, (n-1)*(n+2)*(n-1)!/4) \\ _G. C. Greubel_, Jul 07 2019
%o A138772 (Magma) [(n-1)*(n+2)*Factorial(n-1)/4: n in [1..30]]; // _G. C. Greubel_, Jul 07 2019
%o A138772 (Sage) [(n-1)*(n+2)*factorial(n-1)/4 for n in (1..30)] # _G. C. Greubel_, Jul 07 2019
%o A138772 (GAP) List([1..30], n-> (n-1)*(n+2)*Factorial(n-1)/4); # _G. C. Greubel_, Jul 07 2019
%Y A138772 Cf. A138771.
%Y A138772 Cf. A159324, A000254.
%Y A138772 Column k=2 of A185105.
%K A138772 nonn
%O A138772 1,3
%A A138772 _Emeric Deutsch_, Apr 10 2008