A238340 -id:A238340 - cp's OEIS frontend

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

Wesley Ivan Hurt and Antonio Osorio, Feb 24 2014

			                                             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)

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).

Cf. A235988, A238340, A238702.

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

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 *)

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

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
a(n) = 3*a(n-1) - 3*a(n-2) + 3*a(n-3) - 6*a(n-4) + 6*a(n-5) - 3*a(n-6) + 3*a(n-7) - 3*a(n-8) + a(n-9). - Wesley Ivan Hurt, Jun 19 2024

A238702 Sum of the smallest parts of the partitions of 4n into 4 parts.

Original entry on oeis.org

1, 6, 21, 55, 119, 227, 396, 645, 996, 1474, 2106, 2922, 3955, 5240, 6815, 8721, 11001, 13701, 16870, 20559, 24822, 29716, 35300, 41636, 48789, 56826, 65817, 75835, 86955, 99255, 112816, 127721, 144056, 161910, 181374, 202542, 225511, 250380, 277251, 306229

Offset: 1

Wesley Ivan Hurt and Antonio Osorio, Mar 03 2014

Partial sums of A238340. - Wesley Ivan Hurt, May 27 2014

			Add the numbers 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               6              21              55        ..   a(n)

Vincenzo Librandi, Table of n, a(n) for n = 1..100
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 (4,-6,5,-5,6,-4,1).

Cf. A238328, A238340.

CoefficientList[Series[(x + 1)*(2*x^2 + x + 1)/((1 - x)^5*(x^2 + x + 1)), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 27 2014 *)
LinearRecurrence[{4, -6, 5, -5, 6, -4, 1}, {1, 6, 21, 55, 119, 227, 396}, 50] (* Vincenzo Librandi, Aug 29 2015 *)

Vec(-x*(x+1)*(2*x^2+x+1)/((x-1)^5*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Mar 23 2014

G.f.: -x*(x+1)*(2*x^2+x+1) / ((x-1)^5*(x^2+x+1)). - Colin Barker, Mar 10 2014
a(n) = (1/9)*n^4 + (1/3)*n^3 + (5/18)*n^2 + (1/6)*n + O(1). - Ralf Stephan, May 29 2014
a(n) = Sum_{i=1..n} A238340(i). - Wesley Ivan Hurt, May 29 2014
a(n) = (1/4) * Sum_{i=1..n} A238328(i)/i. - Wesley Ivan Hurt, May 29 2014
Recurrence: Let b(1) = 4, with b(n) = (n/(n-1))*b(n-1) + 4n*Sum_{i=0..2n} (floor((4n-2-i)/2)-i) * (floor((sign((floor((4n-2-i)/2)-i))+2)/2)). Then a(1) = 1, with a(n) = b(n)/(4n) + a(n-1), for n>1. - Wesley Ivan Hurt, Jun 27 2014
E.g.f.: (exp(x)*(4 + 3*x*(16 + x*(37 + 2*x*(9 + x)))) - 4*exp(-x/2)*cos(sqrt(3)*x/2))/54. - Stefano Spezia, Feb 09 2023
a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 5*a(n-4) + 6*a(n-5) - 4*a(n-6) + a(n-7). - Wesley Ivan Hurt, Jun 19 2024

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

Original entry on oeis.org

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

Wesley Ivan Hurt and Antonio Osorio, Mar 03 2014

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

			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)

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).

Cf. A238328, A238340, A238702.

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 *)

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

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

A238706 Sum of the smallest parts of the partitions of 4n into 4 parts with smallest part greater than 1.

Original entry on oeis.org

0, 2, 11, 36, 89, 183, 335, 565, 894, 1347, 1952, 2738, 3738, 4988, 6525, 8390, 10627, 13281, 16401, 20039, 24248, 29085, 34610, 40884, 47972, 55942, 64863, 74808, 85853, 98075, 111555, 126377, 142626, 160391, 179764, 200838, 223710, 248480, 275249, 304122

Offset: 1

Wesley Ivan Hurt and Antonio Osorio, Mar 03 2014

			Add the numbers > 1 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
------------------------------------------------------------------------
     0               2              11              36        ..   a(n)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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 (4,-6,5,-5,6,-4,1).

Cf. A238328, A238340, A238702, A238705.

I:=[0,2,11,36,89,183,335]; [n le 7 select I[n] else 4*Self(n-1)-6*Self(n-2)+5*Self(n-3)-5*Self(n-4)+6*Self(n-5)-4*Self(n-6)+Self(n-7): n in [1..40]]; // Vincenzo Librandi, Mar 24 2014

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); b[0] = 0; c[1] = 1; c[n_] := b[n] + c[n - 1]; Table[c[n] - (b[n] - b[n - 1]), {n, 50}]
CoefficientList[Series[x (x - 2) (x + 1) (2 x^2 + x + 1)/((x - 1)^5 (x^2 + x + 1)), {x, 0, 40}],x] (* Vincenzo Librandi, Mar 24 2014 *)
Table[Total[Select[IntegerPartitions[4n,{4}],#[[-1]]>1&][[All,-1]]],{n,40}] (* or *) LinearRecurrence[{4,-6,5,-5,6,-4,1},{0,2,11,36,89,183,335},40] (* Harvey P. Dale, Jan 06 2023 *)

concat(0, Vec(x^2*(x-2)*(x+1)*(2*x^2+x+1)/((x-1)^5*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Mar 23 2014

G.f.: x^2*(x-2)*(x+1)*(2*x^2+x+1) / ((x-1)^5*(x^2+x+1)). - Colin Barker, Mar 23 2014

a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 5*a(n-4) + 6*a(n-5) - 4*a(n-6) + a(n-7) for n > 7. - Wesley Ivan Hurt, Oct 07 2017

A239056 Sum of the parts in the partitions of 4n into 4 parts with smallest part = 1.

Original entry on oeis.org

4, 32, 120, 304, 600, 1056, 1708, 2560, 3672, 5080, 6776, 8832, 11284, 14112, 17400, 21184, 25432, 30240, 35644, 41600, 48216, 55528, 63480, 72192, 81700, 91936, 103032, 115024, 127832, 141600, 156364, 172032, 188760, 206584, 225400, 245376, 266548, 288800

Offset: 1

Wesley Ivan Hurt, Mar 09 2014

All terms are multiples of 4.

			For a(n) add the parts in the partitions of 4n with smallest part = 1.
                                              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
------------------------------------------------------------------------
     4               32            120             304        ..   a(n)

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

Cf. A238328, A238340, A238702, A238705, A238706.

b[n_] := Sum[((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; Table[b[n], {n, 50}]
LinearRecurrence[{2,-1,2,-4,2,-1,2,-1},{4,32,120,304,600,1056,1708,2560},40] (* Harvey P. Dale, Oct 18 2018 *)

Vec(4*x*(2*x^6+10*x^5+16*x^4+22*x^3+15*x^2+6*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 22 2014

G.f.: 4*x*(2*x^6+10*x^5+16*x^4+22*x^3+15*x^2+6*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 10 2014

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - a(n-6) + 2*a(n-7) - a(n-8). - Wesley Ivan Hurt, Jun 20 2024

A239057 Sum of the parts in the partitions of 4n into 4 parts with smallest part equal to 1 minus the number of these partitions.

Original entry on oeis.org

3, 28, 110, 285, 570, 1012, 1647, 2480, 3570, 4953, 6622, 8648, 11067, 13860, 17110, 20853, 25058, 29820, 35175, 41080, 47642, 54897, 62790, 71440, 80883, 91052, 102078, 113997, 126730, 140420, 155103, 170688, 187330, 205065, 223790, 243672, 264747, 286900

Offset: 1

Wesley Ivan Hurt, Mar 09 2014

nonn

			For a(n) add the numbers in the first 3 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
------------------------------------------------------------------------
     3               28            110             285        ..   a(n)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

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

b[n_] := (4 n - 1) Sum[((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]/(4 n); Table[b[n], {n, 50}]
CoefficientList[Series[(2 x^2 + x + 3) (5 x^4 + 19 x^3 + 16 x^2 + 7 x + 1)/((x^2 + x + 1)^2 (x - 1)^4), {x, 0, 50}], x] (* Vincenzo Librandi, Mar 13 2014 *)

a(n) = A239056(n) - A238705(n).
G.f.: x*(2*x^2+x+3)*(5*x^4+19*x^3+16*x^2+7*x+1)/((x^2+x+1)^2*(x-1)^4). - Alois P. Heinz, Mar 11 2014
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - a(n-6) + 2*a(n-7) - a(n-8). - Wesley Ivan Hurt, Jun 22 2024

A239059 Sum of the two smallest parts from the partitions of 4n into 4 parts with smallest part = 1.

Original entry on oeis.org

2, 9, 27, 61, 108, 178, 276, 395, 549, 743, 966, 1236, 1558, 1917, 2335, 2817, 3344, 3942, 4616, 5343, 6153, 7051, 8010, 9064, 10218, 11441, 12771, 14213, 15732, 17370, 19132, 20979, 22957, 25071, 27278, 29628, 32126, 34725, 37479, 40393, 43416, 46606, 49968

Offset: 1

Wesley Ivan Hurt, Mar 09 2014

			For a(n) add the smallest two parts in the partitions with smallest part equal to 1.
                                             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
------------------------------------------------------------------------
     2               9              27              61        ..   a(n)

Iain Fox, Table of n, a(n) for n = 1..10000 (first 200 terms from Vincenzo Librandi)
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-4,2,-1,2,-1).

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

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

Vec(-x*(x^2+x+2)*(2*x^4-3*x^3-4*x^2-2*x-1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 22 2014

G.f.: -x*(x^2+x+2)*(2*x^4-3*x^3-4*x^2-2*x-1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 10 2014

a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3) - 4*a(n-4) + 2*a(n-5) - a(n-6) + 2*a(n-7) - a(n-8). - Wesley Ivan Hurt, Jun 20 2024

A308867 Sum of all the parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 6, 7, 16, 27, 50, 77, 132, 182, 280, 390, 560, 748, 1044, 1349, 1800, 2310, 2992, 3749, 4776, 5875, 7332, 8937, 10948, 13166, 15960, 18972, 22688, 26763, 31654, 36995, 43416, 50320, 58520, 67431, 77800, 89052, 102144, 116186, 132396, 149895

Offset: 0

Wesley Ivan Hurt, Jun 29 2019

nonn

Index entries for sequences related to partitions

Cf. A238340, A308868, A308869, A306670, A306671, A308872, A308873.

Table[n*Sum[Sum[Sum[Sum[Sum[1, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 100}]
Table[Total[Flatten[IntegerPartitions[n,{6}]]],{n,0,50}] (* Harvey P. Dale, Oct 29 2024 *)

a(n) = n * Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} 1.
a(n) = n * A238340(n).
a(n) = A308868(n) + A308869(n) + A306670(n) + A306671(n) + A308872(n) + A308873(n).

A308868 Sum of the smallest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 15, 22, 29, 40, 51, 70, 86, 112, 139, 176, 214, 269, 321, 394, 470, 567, 668, 801, 933, 1103, 1281, 1498, 1725, 2007, 2293, 2643, 3010, 3443, 3897, 4439, 4995, 5652, 6341, 7135, 7967, 8933, 9930, 11079, 12283, 13645

Offset: 0

Wesley Ivan Hurt, Jun 29 2019

nonn

Index entries for sequences related to partitions

Cf. A238340, A308867, A308869, A306670, A306671, A308872, A308873.

Table[Sum[Sum[Sum[Sum[Sum[m, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

a(n) = n * Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} m.

a(n) = A308867(n) - A308869(n) - A306670(n) - A306671(n) - A308872(n) - A308873(n).

A308869 Sum of the fifth largest parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 13, 17, 25, 34, 48, 63, 86, 109, 143, 182, 232, 288, 363, 442, 547, 662, 804, 961, 1157, 1368, 1626, 1909, 2245, 2613, 3054, 3525, 4082, 4688, 5388, 6150, 7031, 7974, 9059, 10231, 11560, 12991, 14614, 16346, 18300, 20400

Offset: 0

Wesley Ivan Hurt, Jun 29 2019

nonn

Index entries for sequences related to partitions

Cf. A238340, A308867, A308868, A306670, A306671, A308872, A308873.

Table[Sum[Sum[Sum[Sum[Sum[l, {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} l.

a(n) = A308867(n) - A308868(n) - A306670(n) - A306671(n) - A308872(n) - A308873(n).

cp's OEIS Frontend

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

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A238702 Sum of the smallest parts of the partitions of 4n into 4 parts.

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

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A238706 Sum of the smallest parts of the partitions of 4n into 4 parts with smallest part greater than 1.

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A239056 Sum of the parts in the partitions of 4n into 4 parts with smallest part = 1.

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A239057 Sum of the parts in the partitions of 4n into 4 parts with smallest part equal to 1 minus the number of these partitions.

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A239059 Sum of the two smallest parts from the partitions of 4n into 4 parts with smallest part = 1.

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