A102169 a(n) = the number of sequences of n integers such that each integer is in the range 0..4 and the sum of the integers is in the range 0..24.

5, 25, 125, 625, 3125, 15625, 78005, 384550, 1829850, 8209410, 34219650, 131875900, 470597480, 1562441800, 4855374080, 14208711350, 39381411950, 103917328350, 262270328730, 635683810740, 1484963848500, 3353799866500

Offset

1,1

Comments

Changing 4 to 2 and 24 to 3 gives A105163. - Don Reble, Aug 14 2012

Links

Colin Barker, Table of n, a(n) for n = 1..1000

Formula

From Michael David Hirschhorn, Aug 10 2012: (Start)

a(n) is the sum of the coefficients of 1, x, x^2, ..., x^24 in (1+x+x^2+x^3+x^4)^n = (1-x^5)^n/(1-x)^n.

But this is equal to the coefficient of x^24 in (1-x^5)^n/(1-x)^(n+1) = Sum_{k=0..n} (-1)^k binomial(n,k) x^5k times Sum_{m>=0} binomial(n+m,m) x^m.

Hence a(n) = Sum_{k=0..4} (-1)^k binomial(n,k) binomial(n+24-5k,n).

For example, if n=2, a(2) = 325-420+120 = 25. (End)

G.f.: -x*(x^24 -25*x^23 +300*x^22 -2300*x^21 +12650*x^20 -53060*x^19 +175980*x^18 -472300*x^17 +1042375*x^16 -1915575*x^15 +2962780*x^14 -3894200*x^13 +4384980*x^12 -4251000*x^11 +3547700*x^10 -2533840*x^9 +1532975*x^8 -776575*x^7 +325880*x^6 -111900*x^5 +30750*x^4 -6500*x^3 +1000*x^2 -100*x +5) / (x-1)^25. - Colin Barker, Nov 01 2014

Example

a(2)=25 because there are five choices for either integer.

Mathematica

Table[Sum[(-1)^k * Binomial[n, k] * Binomial[n+24-5k, n], {k, 0, 4}], {n, 1, 20}]  (* Vaclav Kotesovec, Nov 01 2014 after Michael David Hirschhorn *)

Crossrefs

Cf. A105163.

Sequence in context: A216126 A335506 A129066 * A060391 A000351 A050735

Adjacent sequences: A102166 A102167 A102168 * A102170 A102171 A102172

Keyword

nonn,easy

Author

Tony Berard (TheMathDude(AT)worldnet.att.net), Feb 16 2005

Extensions

Edited by Don Reble, Mar 19 2007

Status

approved