A238705 Number of partitions of 4n into 4 parts with smallest part = 1.

1, 4, 10, 19, 30, 44, 61, 80, 102, 127, 154, 184, 217, 252, 290, 331, 374, 420, 469, 520, 574, 631, 690, 752, 817, 884, 954, 1027, 1102, 1180, 1261, 1344, 1430, 1519, 1610, 1704, 1801, 1900, 2002, 2107, 2214, 2324, 2437, 2552, 2670, 2791, 2914, 3040, 3169

Offset

1,2

Comments

The number of partitions of 4*(n-1) into at most 3 parts. - Colin Barker, Mar 31 2015

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

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 (2,-1,1,-2,1).

Formula

G.f.: -x*(x+1)*(2*x^2+x+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Mar 10 2014

a(n) = 2*a(n-1)-a(n-2)+a(n-3)-2*a(n-4)+a(n-5). - Wesley Ivan Hurt, Nov 18 2021

Example

Count the 1's in the last 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               4              10              19        ..   a(n)

Mathematica

a[1] = 4; a[n_] := (n/(n - 1))*a[n - 1] + 4 n*Sum[(Floor[(4 n - 2 - i)/2] - i)*(Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; b[n_] := a[n]/(4 n); Table[b[n] - b[n - 1], {n, 50}]
LinearRecurrence[{2, -1, 1, -2, 1}, {1, 4, 10, 19, 30}, 50] (* Harvey P. Dale, Jun 13 2015 *)
Table[Count[IntegerPartitions[4 n, {4}], _?(#[[-1]]==1&)], {n, 50}] (* Harvey P. Dale, Dec 29 2021 *)

Prog

(PARI) Vec(-x*(x+1)*(2*x^2+x+1)/((x-1)^3*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Sep 22 2014

Crossrefs

Cf. A238328, A238340, A238702.

Sequence in context: A050858 A173248 A267882 * A022785 A241239 A057312

Adjacent sequences: A238702 A238703 A238704 * A238706 A238707 A238708

Keyword

nonn,easy

Author

Wesley Ivan Hurt and Antonio Osorio, Mar 03 2014

Status

approved