A191698 - OEIS

%I #18 Sep 08 2022 08:45:57

%S 0,38,204,585,1280,2370,3960,6125,8976,12582,17060,22473,28944,36530,

%T 45360,55485,67040,80070,94716,111017,129120,149058,170984,194925,

%U 221040,249350

%N a(n) = (122n^3 + 140n^2 + 45n + 3n(-1)^n)/8.

%C Let p(n,4) be the number of partitions of n into parts <= 4; then a(n) = p(13n,4) - p(n,4).

%C a(1) = p(13,4) - p(1,4) = 39 - 1 = 38.

%C There are 39 partitions of 13 into parts <= 4:

%C [1,1,1,1,1,1,1,1,1,1,1,1,1]

%C [1,1,1,1,1,1,1,1,1,1,1,2]

%C [1,1,1,1,1,1,1,1,1,1,3], [1,1,1,1,1,1,1,1,1,2,2]

%C [1,1,1,1,1,1,1,1,1,4], [1,1,1,1,1,1,1,1,2,3], [1,1,1,1,1,1,1,2,2,2],

%C [1,1,1,1,1,1,1,2,4], [1,1,1,1,1,1,1,3,3], [1,1,1,1,1,1,2,2,3], [1,1,1,1,1,2,2,2,2]

%C [1,1,1,1,1,1,3,4], [1,1,1,1,1,2,2,4], [1,1,1,1,1,2,3,3], [1,1,1,1,2,2,2,3], [1,1,1,2,2,2,2,2]

%C [1,1,1,1,1,4,4], [1,1,1,1,2,3,4], [1,1,1,1,3,3,3], [1,1,1,2,2,2,4], [1,1,1,2,2,3,3], [1,1,2,2,2,2,3], [1,2,2,2,2,2,2]

%C [1,1,1,2,4,4], [1,1,1,3,3,4], [1,1,2,2,3,4], [1,1,2,3,3,3], [1,2,2,2,2,4], [1,2,2,2,3,3], [2,2,2,3,3,3]

%C [1,1,3,4,4], [1,2,2,4,4], [1,2,3,3,4], [1,3,3,3,3], [2,2,2,3,4], [2,2,3,3,3]

%C [1,4,4,4], [2,3,4,4], [3,3,3,4];

%C and there is 1 partition of 1 into parts < 4:

%C [1].

%H Vincenzo Librandi, <a href="/A191698/b191698.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-4,1,2,-1).

%F G.f.: x*(3*x^4+58*x^3+139*x^2+128*x+38)/((x-1)^4*(x+1)^2). - _Robert Israel_, Dec 09 2016

%p seq((122*n^3 + 140*n^2 + 45*n + 3*n*(-1)^n)/8, n=0..30); # _Robert Israel_, Dec 09 2016

%t Table[1/8*(122n^3 + 140n^2 + 45n + 3n(-1)^n), {n, 0, 25}]

%o (Magma) [1/8*(122*n^3 + 140*n^2 + 45*n + 3*n*(-1)^n): n in [0..35]]; // _Vincenzo Librandi_, Jun 12 2011

%o (PARI) a(n)=((122*n+140)*n+45+3*(-1)^n)*n>>3 \\ _Charles R Greathouse IV_, Jun 12 2011

%K nonn,easy

%O 0,2

%A _Adi Dani_, Jun 12 2011

%E Offset changed from 1 to 0 by _Vincenzo Librandi_, Jun 12 2011