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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. 2
5, 25, 125, 625, 3125, 15625, 78005, 384550, 1829850, 8209410, 34219650, 131875900, 470597480, 1562441800, 4855374080, 14208711350, 39381411950, 103917328350, 262270328730, 635683810740, 1484963848500, 3353799866500 (list; graph; refs; listen; history; text; internal format)
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 Mike 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 Mike Hirschhorn *)

CROSSREFS

Cf. A105163.

Sequence in context: A206451 A216126 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 (djr(AT)nk.ca), Mar 19 2007

STATUS

approved

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Last modified June 24 14:57 EDT 2017. Contains 288697 sequences.