A238328 Sum of all the parts in the partitions of 4n into 4 parts.

4, 40, 180, 544, 1280, 2592, 4732, 7968, 12636, 19120, 27808, 39168, 53716, 71960, 94500, 121984, 155040, 194400, 240844, 295120, 358092, 430672, 513728, 608256, 715300, 835848, 971028, 1122016, 1289920, 1476000, 1681564, 1907840, 2156220, 2428144, 2724960

Offset

1,1

Links

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

Antonio 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).

Formula

Recurrence: a(1) = 4, with a(n) = (n/(n-1))*a(n-1) + 4n*Sum_{i=0..2n} (floor((4n-2-i)/2)-i) * floor((sign(floor((4n-2-i)/2)-i)+2)/2), n > 1.

G.f.: 4*x*(4*x^6+15*x^5+23*x^4+28*x^3+18*x^2+7*x+1) / ((1-x)^5*(x^2+x+1)^2). - Colin Barker, Mar 10 2014

a(n) = 16/9*n^4 + 4/3*n^3 + O(n). - Ralf Stephan, May 29 2014

a(n) = 4n*(A238702(n) - A238702(n-1)), n > 1. - Wesley Ivan Hurt, May 29 2014

a(n) = 4n * A238340(n). - Wesley Ivan Hurt, May 29 2014

E.g.f.: 4*exp(-x/2)*(3*exp(3*x/2)*(8 + x*(37 + x*(27 + 4*x))) + 3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/27. - Stefano Spezia, Feb 09 2023

Example

                                             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
------------------------------------------------------------------------
     4               40            180             544        ..   a(n)

Mathematica

CoefficientList[Series[4*(4*x^6 + 15*x^5 + 23*x^4 + 28*x^3 + 18*x^2 + 7*x + 1)/((1 - x)^5*(x^2 + x + 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 27 2014 *)
LinearRecurrence[{3, -3, 3, -6, 6, -3, 3, -3, 1}, {4, 40, 180, 544, 1280, 2592, 4732, 7968, 12636}, 50] (* Vincenzo Librandi, Aug 29 2015 *)

Prog

(PARI) Vec(-4*x*(4*x^6+15*x^5+23*x^4+28*x^3+18*x^2+7*x+1)/((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Mar 24 2014
(Magma) I:=[4, 40, 180, 544, 1280, 2592, 4732, 7968, 12636]; [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..45]]; // Vincenzo Librandi, Aug 29 2015

Crossrefs

Cf. A235988, A238340, A238702.

Sequence in context: A224086 A271013 A163322 * A009355 A355838 A061132

Adjacent sequences: A238325 A238326 A238327 * A238329 A238330 A238331

Keyword

nonn,easy

Author

Wesley Ivan Hurt and Antonio Osorio, Feb 24 2014

Status

approved