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

Author

Wolfdieter Lang

Keywords

• easy
• nonn

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

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

Crossrefs

Cf. A001787, A038846.

Programs

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 *)

Sage

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

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