A123296 Number of permutations of n distinct letters (ABCD...) each of which appears 5 times and having n-2 fixed points.

0, 25, 75, 150, 250, 375, 525, 700, 900, 1125, 1375, 1650, 1950, 2275, 2625, 3000, 3400, 3825, 4275, 4750, 5250, 5775, 6325, 6900, 7500, 8125, 8775, 9450, 10150, 10875, 11625, 12400, 13200, 14025, 14875, 15750, 16650, 17575, 18525, 19500, 20500, 21525, 22575

Offset: 0

Author

Zerinvary Lajos, Nov 07 2006

Keywords

• nonn
• easy

Comments

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

Examples

			1
0, 0, 0, "0", 0, 1
1, 0, 25, 0, 100, 0, 100, 0, "25", 0, 1
2252, 15150, 48600, 99350, 144150, 156753, 131000, 87075, 45000,
19300, 6000, 1800, 250, "75", 0, 1
44127009, 274314600, 822998550, 1583402400, 2189652825, 2311947008,
1932997200, 1310330400, 731686550, 340071600, 132480756,
43364000, 11973150, 2760000, 541600, 84000, 12225, 1000, "150", 0, 1
etc...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A059062.

Programs

GAP

List([0..40], n-> 25*Binomial(n+1,2)); # G. C. Greubel, Mar 08 2019

Magma

[25*Binomial(n+1,2): n in [0..40]]; // G. C. Greubel, Mar 08 2019

Maple

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;
seq(binomial(n+1,2)*5^2, n=0..44); # Zerinvary Lajos, Aug 09 2008

Mathematica

LinearRecurrence[{3,-3,1}, {0,25,75}, 40] (* G. C. Greubel, Mar 08 2019 *)

PARI

a(n)=25*n*(n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

Sage

[25*binomial(n+1,2) for n in (0..40)] # G. C. Greubel, Mar 08 2019

Formula

From Zerinvary Lajos, Aug 09 2008: (Start)

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

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

a(n) = 25*n + a(n-1) (with a(0)=0). - Vincenzo Librandi, Nov 13 2010

E.g.f.: (25/2)*x*(2+x)*exp(x). - G. C. Greubel, Mar 08 2019

From Amiram Eldar, Sep 04 2022: (Start)

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

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