A239057 Sum of the parts in the partitions of 4n into 4 parts with smallest part equal to 1 minus the number of these partitions.

3, 28, 110, 285, 570, 1012, 1647, 2480, 3570, 4953, 6622, 8648, 11067, 13860, 17110, 20853, 25058, 29820, 35175, 41080, 47642, 54897, 62790, 71440, 80883, 91052, 102078, 113997, 126730, 140420, 155103, 170688, 187330, 205065, 223790, 243672, 264747, 286900

Offset: 1

Author

Wesley Ivan Hurt, Mar 09 2014

Keywords

• nonn

Examples

			For a(n) add the numbers in the first 3 columns.
                                               13 + 1 + 1 + 1
                                               12 + 2 + 1 + 1
                                               11 + 3 + 1 + 1
                                               10 + 4 + 1 + 1
                                                9 + 5 + 1 + 1
                                                8 + 6 + 1 + 1
                                                7 + 7 + 1 + 1
                                               11 + 2 + 2 + 1
                                               10 + 3 + 2 + 1
                               9 + 1 + 1 + 1    9 + 4 + 2 + 1
                               8 + 2 + 1 + 1    8 + 5 + 2 + 1
                               7 + 3 + 1 + 1    7 + 6 + 2 + 1
                               6 + 4 + 1 + 1    9 + 3 + 3 + 1
                               5 + 5 + 1 + 1    8 + 4 + 3 + 1
                               7 + 2 + 2 + 1    7 + 5 + 3 + 1
               5 + 1 + 1 + 1   6 + 3 + 2 + 1    6 + 6 + 3 + 1
               4 + 2 + 1 + 1   5 + 4 + 2 + 1    7 + 4 + 4 + 1
               3 + 3 + 1 + 1   5 + 3 + 3 + 1    6 + 5 + 4 + 1
1 + 1 + 1 + 1  3 + 2 + 2 + 1   4 + 4 + 3 + 1    5 + 5 + 5 + 1
    4(1)            4(2)           4(3)            4(4)       ..   4n
------------------------------------------------------------------------
     3               28            110             285        ..   a(n)
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Crossrefs

Cf. A238328, A238340, A238702, A238705, A238706, A239056.

Programs

Mathematica

b[n_] := (4 n - 1) Sum[((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]/(4 n); Table[b[n], {n, 50}]
CoefficientList[Series[(2 x^2 + x + 3) (5 x^4 + 19 x^3 + 16 x^2 + 7 x + 1)/((x^2 + x + 1)^2 (x - 1)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 13 2014 *)

Formula

a(n) = A239056(n) - A238705(n).

G.f.: x*(2*x^2+x+3)*(5*x^4+19*x^3+16*x^2+7*x+1)/((x^2+x+1)^2*(x-1)^4). - Alois P. Heinz, Mar 11 2014

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - a(n-6) + 2*a(n-7) - a(n-8). - Wesley Ivan Hurt, Jun 22 2024