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Expansion of 1/((1-x^3)*(1-x^5)*(1-x^7)).
2

%I #57 Sep 08 2022 08:44:36

%S 1,0,0,1,0,1,1,1,1,1,2,1,2,2,2,3,2,3,3,3,4,4,4,4,5,5,5,6,6,6,7,7,7,8,

%T 8,9,9,9,10,10,11,11,12,12,12,14,13,14,15,15,16,16,17,17,18,19,19,20,

%U 20,21,22,22,23,24,24,25,26,26,27,28,29,29,30,31,31,33,33,34,35,35,37,37,38,39,40,41,41,43

%N Expansion of 1/((1-x^3)*(1-x^5)*(1-x^7)).

%C Number of partitions of n into parts 3, 5, and 7. - _Joerg Arndt_, Aug 17 2013

%C Number of different total numbers of kicks, tries and converted tries which lead to a score of n in a rugby (union) match. - _Matthew Scroggs_, Jul 09 2015

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 114, [6t].

%H Vincenzo Librandi, <a href="/A008677/b008677.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=230">Encyclopedia of Combinatorial Structures 230</a>

%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

%H <a href="/index/Rec#order_15">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,0,1,0,1,-1,0,-1,0,-1,0,0,1).

%F a(n) = a(n-3) + a(n-5) + a(n-7) - a(n-8) - a(n-10) - a(n-12) + a(n-15) for n >= 15. - _David Neil McGrath_, Sep 03 2014

%F G.f.: 1 / ((1 - x^3) * (1 - x^5) * (1 - x^7)).

%F Euler transform of length 7 sequence [ 0, 0, 1, 0, 1, 0, 1]. - _Michael Somos_, Sep 30 2014

%F a(n) = a(-15-n) for all n in Z. - _Michael Somos_, Sep 30 2014

%F 0 = a(n) - a(n+3) - a(n+5) + a(n+8) - [mod(n, 7) == 6] for all n in Z. - _Michael Somos_, Sep 30 2014

%F a(n) = round(n^2/210 + n/14 + 5/21) + r(n) where r(n) = 1 if n == 0, 3, 10, 15, 45, 75, 80, 87, or 90 (mod 105), r(n) = -1 if n == 4, 11, 16, 44, 46, 74, 79 or 86 (mod 105), r(n) = 0 otherwise. - _Robert Israel_, Jul 09 2015

%e G.f. = 1 + x^3 + x^5 + x^6 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + 2*x^12 + ...

%p S:= series(1/((1-x^3)*(1-x^5)*(1-x^7)), x, 101):

%p seq(coeff(S,x,j),j=0..100); # _Robert Israel_, Jul 09 2015

%t CoefficientList[Series[1/((1-x^3)(1-x^5)(1-x^7)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Jun 23 2013 *)

%t LinearRecurrence[{0,0,1,0,1,0,1,-1,0,-1,0,-1,0,0,1},{1,0,0,1,0,1,1,1,1,1,2,1,2,2,2},70] (* _Harvey P. Dale_, Aug 04 2020 *)

%o (PARI) a(n)=[1,0,-2,2,-2,0,1][n%7+1]/7 +[2,-1,0,0,-1][n%5+1]/5 +[2,-1,-1][n%3+1]/9 +(3*n^2+45*n+148)/630; \\ _Tani Akinari_, Aug 17 2013

%o (PARI) a(n)=floor((n^2+15*n+86)/210+(n%3<1)/3+3*(n%5<1)/5) \\ _Tani Akinari_, Sep 30 2014

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 100); Coefficients(R!( 1/((1-x^3)*(1-x^5)*(1-x^7)) )); // _G. C. Greubel_, Sep 08 2019

%o (Sage)

%o def A008677_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P(1/((1-x^3)*(1-x^5)*(1-x^7))).list()

%o A008677_list(100) # _G. C. Greubel_, Sep 08 2019

%K nonn,easy

%O 0,11

%A _N. J. A. Sloane_

%E Typo in name fixed by _Vincenzo Librandi_, Jun 23 2013