A240707 Sum of the middle parts in the partitions of 4n-1 into 3 parts.

1, 8, 31, 80, 159, 282, 459, 690, 993, 1378, 1841, 2404, 3077, 3852, 4755, 5796, 6963, 8286, 9775, 11414, 13237, 15254, 17445, 19848, 22473, 25296, 28359, 31672, 35207, 39010, 43091, 47418, 52041, 56970, 62169, 67692, 73549, 79700, 86203, 93068, 100251

Offset

1,2

Comments

Original definition: Sum of the second largest parts in the partitions of 4n into 4 parts with smallest part = 1 (see the example).

Links

Table of n, a(n) for n=1..41.

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*(x^2+3*x+1)*(3*x^4+3*x^3+6*x^2+3*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Apr 13 2014

Example

For a(n) add the parts in the second 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               8              31              80        ..   a(n)

Maple

A240707:=n->add(add(i*floor((signum((floor((4*n-2-j)/2)-j))+2)/2), i=j+1..floor((4*n-2-j)/2)), j=0..2*n); seq(A240707(n), n=1..50);

Mathematica

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[c[n], {n, 50}]

Prog

(PARI) Vec(x*(x^2+3*x+1)*(3*x^4+3*x^3+6*x^2+3*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Apr 13 2014
(PARI) A240707(n)=sum(a=1, (4*n-1)\3, (4*n-1-a)\2*((4*n-1-a)\2+1)-a*(a-1))\2 \\ The summand is sum(b=a, (4*n-1-a)\2, b). - M. F. Hasler, Apr 17 2014

Crossrefs

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

Sequence in context: A022273 A302874 A061294 * A115293 A212579 A115004

Adjacent sequences: A240704 A240705 A240706 * A240708 A240709 A240710

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 10 2014

Extensions

Definition simplified by M. F. Hasler, Apr 17 2014

Status

approved