cp's OEIS Frontend

A000115 Denumerants: Expansion of 1/((1-x)*(1-x^2)*(1-x^5)).

%I A000115 M0279 N0098 #61 Apr 30 2025 10:08:56
%S A000115 1,1,2,2,3,4,5,6,7,8,10,11,13,14,16,18,20,22,24,26,29,31,34,36,39,42,
%T A000115 45,48,51,54,58,61,65,68,72,76,80,84,88,92,97,101,106,110,115,120,125,
%U A000115 130,135,140,146,151,157,162,168,174,180,186,192,198,205,211,218,224,231,238
%N A000115 Denumerants: Expansion of 1/((1-x)*(1-x^2)*(1-x^5)).
%C A000115 Number of partitions of n into parts 1, 2, or 5.
%C A000115 First differences are in A008616. First differences of A001304. Pairwise sums of A008720.
%D A000115 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, D(n;1,2,5).
%D A000115 M. Jeger, Ein partitions problem ..., Elemente de Math., 13 (1958), 97-120.
%D A000115 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 152.
%D A000115 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000115 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000115 Vincenzo Librandi, <a href="/A000115/b000115.txt">Table of n, a(n) for n = 0..10000</a>
%H A000115 Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Ramirez/ramirez19.html">Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 15.
%H A000115 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1,0,1,-1,-1,1)
%F A000115 a(n) = round((n+4)^2/20).
%F A000115 a(n) = a(-8 - n) for all n in Z. - _Michael Somos_, May 28 2014
%e A000115 G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + ...
%p A000115 1/((1-x)*(1-x^2)*(1-x^5)): seq(coeff(series(%, x, n+1), x, n), n=0..65);
%p A000115 # next Maple program:
%p A000115 s:=proc(n) if n mod 5 = 0 then RETURN(1); fi; if n mod 5 = 1 then RETURN(0); fi; if n mod 5 = 2 then RETURN(1); fi; if n mod 5 = 3 then RETURN(-1); fi; if n mod 5 = 4 then RETURN(-1); fi; end: f:=n->(2*n^2+16*n+27+5*(-1)^n+8*s(n))/40: seq(f(n), n=0..65);  # from Jeger's paper
%t A000115 nn=50;CoefficientList[Series[1/(1-x)/(1-x^2)/(1-x^5),{x,0,nn}],x]  (* _Geoffrey Critzer_, Jan 20 2013 *)
%t A000115 LinearRecurrence[{1,1,-1,0,1,-1,-1,1},{1,1,2,2,3,4,5,6},70] (* _Harvey P. Dale_, Sep 27 2019 *)
%o A000115 (Magma) [Round((n+4)^2/20): n in [0..70]]; // _Vincenzo Librandi_, Jun 23 2011
%o A000115 (PARI) a(n)=(n^2+8*n+26)\20 \\ _Charles R Greathouse IV_, Jun 23 2011
%Y A000115 Cf. A001304, A008616, A008720.
%K A000115 nonn,easy
%O A000115 0,3
%A A000115 _N. J. A. Sloane_