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

Author

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

Keywords

• nonn
• easy

Comments

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

Examples

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

Links

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

Crossrefs

Cf. A105163.

Programs

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 *)

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

Extensions

Edited by Don Reble, Mar 19 2007