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%I #40 Oct 30 2022 09:00:12
%S 1,0,0,1,1,1,2,3,3,5,8,10,14,21,29,40,58,82,114,162,230,323,456,646,
%T 911,1285,1817,2566,3621,5115,7225,10200,14404,20344,28727,40565,
%U 57288,80900,114240,161328,227824,321720,454321,641580,906012,1279433,1806773,2551457,3603066,5088119,7185255,10146741,14328848
%N Expansion of 1/(1-x^3-x^4-x^5-x^6-x^7).
%C Number of compositions of n into parts 3, 4, 5, 6 and 7 - _David Neil McGrath_, Aug 17 2014
%H Harvey P. Dale, <a href="/A017820/b017820.txt">Table of n, a(n) for n = 0..1000</a>
%H Vladimir Kruchinin and V. D. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1,1,1,1).
%F a(n) = sum(sum(C(k,r)*sum(C(r,m)*sum(C(m,j)*C(j,n-m-3*k-j-r), j=0..m), m=0..r), r=0..k), k=1..n), n>0. - _Vladimir Kruchinin_, Aug 30 2010
%F a(0)=1, a(1)=0, a(2)=0, a(3)=1, a(4)=1, a(5)=1, a(6)=2; for n>6, a(n) = a(n-3)+a(n-4)+ a(n-5)+a(n-6)+a(n-7). - _Harvey P. Dale_, May 11 2012
%t CoefficientList[Series[1/(1 - x^3 - x^4 - x^5 - x^6 - x^7), {x, 0, 50}], x] (* or *) LinearRecurrence[{0, 0, 1, 1 , 1, 1, 1}, {1, 0, 0, 1, 1, 1, 2}, 50] (* _Harvey P. Dale_, May 11 2012 *)
%t CoefficientList[Series[1 / (1 - Total[x^Range[3, 7]]), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 27 2013 *)
%o (Maxima) a(n):=sum(sum(binomial(k,r) *sum(binomial(r,m)*sum(binomial(m,j) *binomial(j,n-m-3*k-j-r), j,0,m), m,0,r), r,0,k), k,1,n); /* _Vladimir Kruchinin_, Aug 30 2010 */
%o (PARI) Vec(1/(1-x^3-x^4-x^5-x^6-x^7)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%o (Magma) I:=[1,0,0,1,1,1,2]; [n le 7 select I[n] else Self(n-3)+Self(n-4) +Self(n-5)+Self(n-6)+Self(n-7): n in [1..50]]; /* or */ m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/(1-x^3-x^4-x^5-x^6-x^7))); // _Vincenzo Librandi_, Jun 27 2013
%K nonn,easy
%O 0,7
%A _N. J. A. Sloane_.
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