A240711 Sum of the largest parts in the partitions of 4n into 4 parts with smallest part = 1.

1, 15, 62, 163, 333, 596, 973, 1475, 2130, 2959, 3969, 5192, 6649, 8343, 10310, 12571, 15125, 18012, 21253, 24843, 28826, 33223, 38025, 43280, 49009, 55199, 61902, 69139, 76893, 85220, 94141, 103635, 113762, 124543, 135953, 148056, 160873, 174375, 188630

Offset

1,2

Links

Table of n, a(n) for n=1..39.

Index entries for sequences related to partitions

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

Formula

G.f.: x*(7*x^6+27*x^5+43*x^4+52*x^3+33*x^2+13*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Apr 11 2014

Example

For a(n) add the parts in the first 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
------------------------------------------------------------------------
     1               15             62             163       ..   a(n)

Mathematica

b[n_] := Sum[((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; c[1] = 1; c[n_] := Sum[Sum[i (Floor[(Sign[(Floor[(4 n - 2 - j)/2] - j)] + 2)/2]), {i, j + 1, Floor[(4 n - 2 - j)/2]}], {j, 0, 2 n}]; Table[b[n] - c[n], {n, 50}]

Prog

(PARI) Vec(x*(7*x^6+27*x^5+43*x^4+52*x^3+33*x^2+13*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Apr 11 2014

Crossrefs

Cf. A238328, A238340, A238702, A238705, A238706, A239056, A239057, A239059, A240707.

Sequence in context: A219819 A157446 A220084 * A212055 A219296 A067572

Adjacent sequences: A240708 A240709 A240710 * A240712 A240713 A240714

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 10 2014

Status

approved