0,2

With a different offset, number of n-permutations (n = 5) of 6 objects u, v, w, z, x, y with repetition allowed, containing exactly 4 u's. Example: a(1) = 25 because we have uuuuv, uuuvu, uuvuu, uvuuu, vuuuu, uuuuw, uuuwu, uuwuu, uwuuu, wuuuu, uuuuz, uuuzu, uuzuu, uzuuu, zuuuu, uuuux, uuuxu, uuxuu, uxuuu, xuuuu uuuuy, uuuyu, uuyuu, uyuuu, yuuuu. - Zerinvary Lajos, Jun 12 2008

Index entries for linear recurrences with constant coefficients, signature (25,-250,1250,-3125,3125).

a(n) = binomial(n+4, 4)*5^n.

G.f.: 1/(1-5*x)^5.

a(n) = 25*a(n-1) - 250*a(n-2) + 1250*a(n-3) - 3125*a(n-4) + 3125*a(n-5), a(0)=1, a(1)=25, a(2)=375, a(3)=4375, a(4)=43750. - Harvey P. Dale, Mar 20 2013

From Amiram Eldar, Nov 03 2025: (Start)

Sum_{n>=0} 1/a(n) = 860/3 - 1280*log(5/4).

Sum_{n>=0} (-1)^n/a(n) = 4320*log(6/5) - 2360/3. (End)

seq(binomial(n+4, 4)*5^n, n=0..18); # Zerinvary Lajos, Jun 12 2008

CoefficientList[Series[1/(1-5x)^5, {x, 0, 30}], x] (* or *) LinearRecurrence[ {25, -250, 1250, -3125, 3125}, {1, 25, 375, 4375, 43750}, 30] (* Harvey P. Dale, Mar 20 2013 *)

(SageMath) [lucas_number2(n, 5, 0)*binomial(n, 4)/5^4 for n in range(4, 23)] # Zerinvary Lajos, Mar 12 2009

Cf. A001787, A038846.

Sequence in context: A227024 A254376 A022749 * A225968 A094190 A069396

Adjacent sequences: A036068 A036069 A036070 * A036072 A036073 A036074

easy,nonn

approved