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A123296 Number of permutations of n distinct letters (ABCD...) each of which appears 5 times and having n-2 fixed points. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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).

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

EXAMPLE

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...

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 *)

PROG

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

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

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

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

CROSSREFS

Cf. A059062.

Sequence in context: A045180 A270385 A053742 * A118610 A008852 A042226

Adjacent sequences:  A123293 A123294 A123295 * A123297 A123298 A123299

KEYWORD

nonn,easy

AUTHOR

Zerinvary Lajos, Nov 07 2006

STATUS

approved

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Last modified February 28 06:55 EST 2020. Contains 332321 sequences. (Running on oeis4.)