A123296 - OEIS

%I #26 Sep 04 2022 04:09:45

%S 0,25,75,150,250,375,525,700,900,1125,1375,1650,1950,2275,2625,3000,

%T 3400,3825,4275,4750,5250,5775,6325,6900,7500,8125,8775,9450,10150,

%U 10875,11625,12400,13200,14025,14875,15750,16650,17575,18525,19500,20500,21525,22575

%N Number of permutations of n distinct letters (ABCD...) each of which appears 5 times and having n-2 fixed points.

%C Number of n permutations (n>=2) of 6 objects t, u, v, z, x, y with repetition allowed, containing n-2 u's. Example: if n=2 then n-2=zero (0) u, a(1)=25 because we have tt, tv, tz, tx, ty, vt, vv, vz, vx, vy, zt, zv, zz, zx, zy, xt, xv, xz, xx, xy, yt, yv, yz, yx, yy. if n=3 then n-2=one (1) u, a(2)= 75, if n=4 then n-2=two (2) u, a(2)= 150, if n=5 then n-2=three (3) u a(3)= 250, etc. - _Zerinvary Lajos_, Aug 09 2008

%H G. C. Greubel, <a href="/A123296/b123296.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F From _Zerinvary Lajos_, Aug 09 2008: (Start)

%F G.f.: 25*x/(1-x)^3.

%F a(n) = C(n+1,2)*5^2 = 25*A000217(n), n >= 0. (End)

%F a(n) = 25*n + a(n-1) (with a(0)=0). - _Vincenzo Librandi_, Nov 13 2010

%F E.g.f.: (25/2)*x*(2+x)*exp(x). - _G. C. Greubel_, Mar 08 2019

%F From _Amiram Eldar_, Sep 04 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 2/25.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*(2*log(2)-1)/25. (End)

%e 1

%e 0, 0, 0, "0", 0, 1

%e 1, 0, 25, 0, 100, 0, 100, 0, "25", 0, 1

%e 2252, 15150, 48600, 99350, 144150, 156753, 131000, 87075, 45000,

%e 19300, 6000, 1800, 250, "75", 0, 1

%e 44127009, 274314600, 822998550, 1583402400, 2189652825, 2311947008,

%e 1932997200, 1310330400, 731686550, 340071600, 132480756,

%e 43364000, 11973150, 2760000, 541600, 84000, 12225, 1000, "150", 0, 1

%e etc...

%p p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); for n from 0 to 8 do seq(coeff(f(t, n, 5), t, m)/5!^n, m=0..5*n); od;

%p seq(binomial(n+1,2)*5^2, n=0..44); # _Zerinvary Lajos_, Aug 09 2008

%t LinearRecurrence[{3,-3,1}, {0,25,75}, 40] (* _G. C. Greubel_, Mar 08 2019 *)

%o (PARI) a(n)=25*n*(n+1)/2 \\ _Charles R Greathouse IV_, Jun 17 2017

%o (Magma) [25*Binomial(n+1,2): n in [0..40]]; // _G. C. Greubel_, Mar 08 2019

%o (Sage) [25*binomial(n+1,2) for n in (0..40)] # _G. C. Greubel_, Mar 08 2019

%o (GAP) List([0..40], n-> 25*Binomial(n+1,2)) # _G. C. Greubel_, Mar 08 2019

%Y Cf. A059062.

%K nonn,easy

%O 0,2

%A _Zerinvary Lajos_, Nov 07 2006