A242727 Sum of the third largest parts of the partitions of 4n into 4 parts.

1, 7, 29, 86, 198, 396, 719, 1203, 1899, 2866, 4156, 5840, 7997, 10695, 14025, 18086, 22962, 28764, 35611, 43603, 52871, 63554, 75768, 89664, 105401, 123111, 142965, 165142, 189790, 217100, 247271, 280467, 316899, 356786, 400308, 447696, 499189, 554983

Offset: 1

Author

Wesley Ivan Hurt and Antonio Osorio, May 21 2014

Keywords

• nonn
• easy

Examples

			Add the numbers in the third column for a(n):
                                              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 + 4 + 2 + 1
                                              8 + 5 + 2 + 1
                                              7 + 6 + 2 + 1
                                              9 + 3 + 3 + 1
                                              8 + 4 + 3 + 1
                                              7 + 5 + 3 + 1
                                              6 + 6 + 3 + 1
                                              7 + 4 + 4 + 1
                                              6 + 5 + 4 + 1
                                              5 + 5 + 5 + 1
                              9 + 1 + 1 + 1   10+ 2 + 2 + 2
                              8 + 2 + 1 + 1   9 + 3 + 2 + 2
                              7 + 3 + 1 + 1   8 + 4 + 2 + 2
                              6 + 4 + 1 + 1   7 + 5 + 2 + 2
                              5 + 5 + 1 + 1   6 + 6 + 2 + 2
                              7 + 2 + 2 + 1   8 + 3 + 3 + 2
                              6 + 3 + 2 + 1   7 + 4 + 3 + 2
                              5 + 4 + 2 + 1   6 + 5 + 3 + 2
                              5 + 3 + 3 + 1   6 + 4 + 4 + 2
                              4 + 4 + 3 + 1   5 + 5 + 4 + 2
               5 + 1 + 1 + 1  6 + 2 + 2 + 2   7 + 3 + 3 + 3
               4 + 2 + 1 + 1  5 + 3 + 2 + 2   6 + 4 + 3 + 3
               3 + 3 + 1 + 1  4 + 4 + 2 + 2   5 + 5 + 3 + 3
               3 + 2 + 2 + 1  4 + 3 + 3 + 2   5 + 4 + 4 + 3
1 + 1 + 1 + 1  2 + 2 + 2 + 2  3 + 3 + 3 + 3   4 + 4 + 4 + 4
    4(1)            4(2)           4(3)            4(4)       ..   4n
------------------------------------------------------------------------
     1               7              29              86        ..   a(n)
		

Links

• A. Osorio, A Sequential Allocation Problem: The Asymptotic Distribution of Resources, Munich Personal RePEc Archive, 2014.
• Index entries for sequences related to partitions
• Index entries for linear recurrences with constant coefficients, signature (3,-3,3,-6,6,-3,3,-3,1).

Crossrefs

Cf. A238328, A238340, A238702, A239667, A241084.

Programs

Magma

I:=[1,7,29,86,198,396,719,1203,1899]; [n le 9 select I[n] else 3*Self(n-1)-3*Self(n-2)+3*Self(n-3)-6*Self(n-4)+6*Self(n-5)-3*Self(n-6)+3*Self(n-7)-3*Self(n-8)+Self(n-9): n in [1..40]]; // Vincenzo Librandi, Aug 29 2015

Mathematica

CoefficientList[Series[-(1 + 4x + 11x^2 + 17x^3 + 12x^4 + 9x^5 + 2x^6) / ((-1 + x)^5 (1 + x + x^2)^2), {x, 0, 50}], x]
LinearRecurrence[{3, -3, 3, -6, 6, -3, 3, -3, 1}, {1, 7, 29, 86, 198, 396, 719, 1203, 1899}, 50] (* Vincenzo Librandi, Aug 29 2015 *)

Formula

G.f.: (1 + 4*x + 11*x^2 + 17*x^3 + 12*x^4 + 9*x^5 + 2*x^6) / ((1 - x)^5*(1 + x + x^2)^2).

a(n) = A238328(n) - A239667(n) - A241084(n) - A238702(n).

a(n) = 7/27*n^4 + 35/27*n^3 + 22/9*n^2 + 59/27*n + O(1). - Ralf Stephan, May 26 2014