A191698 a(n) = (122n^3 + 140n^2 + 45n + 3n(-1)^n)/8.

0, 38, 204, 585, 1280, 2370, 3960, 6125, 8976, 12582, 17060, 22473, 28944, 36530, 45360, 55485, 67040, 80070, 94716, 111017, 129120, 149058, 170984, 194925, 221040, 249350

Offset

0,2

Comments

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

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

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

[1,1,1,1,1,1,1,1,1,1,1,1,1]

[1,1,1,1,1,1,1,1,1,1,1,2]

[1,1,1,1,1,1,1,1,1,1,3], [1,1,1,1,1,1,1,1,1,2,2]

[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],

[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]

[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]

[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]

[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]

[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]

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

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

[1].

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Formula

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

Maple

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

Mathematica

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

Prog

(Magma) [1/8*(122*n^3 + 140*n^2 + 45*n + 3*n*(-1)^n): n in [0..35]]; // Vincenzo Librandi, Jun 12 2011
(PARI) a(n)=((122*n+140)*n+45+3*(-1)^n)*n>>3 \\ Charles R Greathouse IV, Jun 12 2011

Crossrefs

Sequence in context: A235079 A005910 A281769 * A124238 A020870 A211502

Adjacent sequences: A191695 A191696 A191697 * A191699 A191700 A191701

Keyword

nonn,easy

Author

Adi Dani, Jun 12 2011

Extensions

Offset changed from 1 to 0 by Vincenzo Librandi, Jun 12 2011

Status

approved