A008673 - OEIS

%I #52 Sep 08 2022 08:44:36

%S 1,1,1,2,2,3,4,5,6,7,9,10,12,14,16,19,21,24,27,30,34,38,42,46,51,56,

%T 61,67,73,79,86,93,100,108,116,125,134,143,153,163,174,185,197,209,

%U 221,235,248,262,277,292,308,324,341,358,376,395,414,434,454,475,497,519,542,566,590

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

%C Number of partitions of n into parts 1, 3, 5, and 7. - _Joerg Arndt_, Jul 08 2013

%C Number of partitions (d1,d2,d3,d4) of n such that 0 <= d1/1 <= d2/2 <= d3/3 <= d4/4. - _Seiichi Manyama_, Jun 04 2017

%H Seiichi Manyama, <a href="/A008673/b008673.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Vincenzo Librandi)

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=234">Encyclopedia of Combinatorial Structures 234</a>

%H <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1,1,-1,1,-2,1,-1,1,-1,1,0,1,-1).

%F a(n) = floor((n^3 + 24*n^2 + 171*n + 630)/630). - _Tani Akinari_, Jul 08 2013

%F a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7) - 2*a(n-8) + a(n-9) - a(n-10) + a(n-11) - a(n-12) + a(n-13) + a(n-15) - a(n-16). - _David Neil McGrath_, Feb 14 2015

%e There are a(7)=5 partitions of n=7 into parts 1, 3, 5, and 7: (7), (511), (331), (31111), and (1111111). - _David Neil McGrath_, Feb 14 2015

%p seq(coeff(series(1/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)), x, n+1), x, n), n = 0 .. 70); # _G. C. Greubel_, Sep 08 2019

%t CoefficientList[Series[1/((1-x)(1-x^3)(1-x^5)(1-x^7)), {x,0,70}], x] (* _Vincenzo Librandi_, Jun 22 2013 *)

%t LinearRecurrence[{1,0,1,-1,1,-1,1,-2,1,-1,1,-1,1,0,1,-1}, {1,1,1,2,2,3, 4,5,6,7,9,10,12,14,16,19}, 70] (* _Harvey P. Dale_, Jul 08 2019 *)

%o (PARI) vector(70, n, m=n-1; (m^3+24*m^2+171*m+630)\630 ) \\ _G. C. Greubel_, Sep 08 2019

%o (Magma) [Floor((n^3+24*n^2+171*n+630)/630): n in [0..70]]; // _G. C. Greubel_, Sep 08 2019

%o (Sage) [floor((n^3+24*n^2+171*n+630)/630) for n in (0..70)] # _G. C. Greubel_, Sep 08 2019

%o (GAP) List([0..70], n-> Int((n^3+24*n^2+171*n+630)/630) ); # _G. C. Greubel_, Sep 08 2019

%Y Cf. A259094.

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_