A036071 Expansion of 1/(1-5*x)^5.

1, 25, 375, 4375, 43750, 393750, 3281250, 25781250, 193359375, 1396484375, 9775390625, 66650390625, 444335937500, 2905273437500, 18676757812500, 118286132812500, 739288330078125, 4566192626953125, 27904510498046875

Offset

0,2

Comments

With a different offset, number of n-permutations (n=5) of 6 objects u, v, w, z, x, y with repetition allowed, containing exactly four (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

Links

Table of n, a(n) for n=0..18.

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

Formula

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

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

Cf. A001787, A038846.

Sequence in context: A227024 A254376 A022749 * A225968 A094190 A069396

Adjacent sequences: A036068 A036069 A036070 * A036072 A036073 A036074

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved