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

1,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

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

Example

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)

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 *)

Crossrefs

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

Sequence in context: A186429 A165393 A107651 * A338791 A100019 A053132

Adjacent sequences: A239054 A239055 A239056 * A239058 A239059 A239060

Keyword

nonn

Author

Wesley Ivan Hurt, Mar 09 2014

Status

approved