A239056 -id:A239056 - cp's OEIS frontend

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.

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

A239186 Sum of the largest two parts in the partitions of 4n into 4 parts with smallest part equal to 1.

Original entry on oeis.org

2, 23, 93, 243, 492, 878, 1432, 2165, 3123, 4337, 5810, 7596, 9726, 12195, 15065, 18367, 22088, 26298, 31028, 36257, 42063, 48477, 55470, 63128, 71482, 80495, 90261, 100811, 112100, 124230, 137232, 151053, 165803, 181513, 198122, 215748, 234422, 254075

Offset: 1

Wesley Ivan Hurt, Mar 11 2014

			For a(n) add the numbers in the first two 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
------------------------------------------------------------------------
     2               23             93             243        ..   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, A239057, A239059.

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

Vec(x*(10*x^6+39*x^5+61*x^4+76*x^3+49*x^2+19*x+2)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 22 2014

G.f.: x*(10*x^6+39*x^5+61*x^4+76*x^3+49*x^2+19*x+2) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 12 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, Nov 19 2021

A239667 Sum of the largest parts of the partitions of 4n into 4 parts.

Original entry on oeis.org

1, 17, 84, 262, 629, 1289, 2370, 4014, 6393, 9703, 14150, 19974, 27439, 36815, 48410, 62556, 79587, 99879, 123832, 151844, 184359, 221845, 264764, 313628, 368973, 431325, 501264, 579394, 666305, 762645, 869086, 986282, 1114949, 1255827, 1409634, 1577154, 1759195, 1956539, 2170038, 2400568

Offset: 1

Wesley Ivan Hurt and Antonio Osorio, Mar 23 2014

			Add the numbers in the first 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               17             84             262        ..   a(n)

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).

Cf. A238328, A238340, A238702, A238705, A238706, A239056, A239057, A239059, A239186.

I:=[1,17,84,262,629,1289,2370,4014,6393]; [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[-(9*x^6 + 32*x^5 + 50*x^4 + 58*x^3 + 36*x^2 + 14*x +
1)/((x - 1)^5*(x^2 + x + 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 13 2014 *)
LinearRecurrence[{3, -3, 3, -6, 6, -3, 3, -3, 1}, {1, 17, 84, 262, 629, 1289, 2370, 4014, 6393}, 50](* Vincenzo Librandi, Aug 29 2015 *)
Table[Total[IntegerPartitions[4 n,{4}][[All,1]]],{n,40}] (* Harvey P. Dale, Apr 25 2020 *)

Vec(-x*(9*x^6+32*x^5+50*x^4+58*x^3+36*x^2+14*x+1) / ((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Mar 23 2014

G.f.: -x*(9*x^6+32*x^5+50*x^4+58*x^3+36*x^2+14*x+1) / ((x-1)^5*(x^2+x+1)^2). - Colin Barker, Mar 23 2014

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) = a(n-1) + b(n-1)/(4n-4) + Sum_{i=j+1..floor((4n-2-j)/2)} ( Sum_{j=0..2n} (4n-2-i-j) * floor((sign(floor((4n-2-j)/2)-j)+2)/2) ). - Wesley Ivan Hurt, Jun 13 2014

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

Original entry on oeis.org

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

Wesley Ivan Hurt, Apr 10 2014

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

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, A239059.

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);

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}]

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

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

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

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

A241084 Sum of the second largest parts of the partitions of 4n into 4 parts.

Original entry on oeis.org

1, 10, 46, 141, 334, 680, 1247, 2106, 3348, 5077, 7396, 10432, 14325, 19210, 25250, 32621, 41490, 52056, 64531, 79114, 96040, 115557, 137896, 163328, 192137, 224586, 260982, 301645, 346870, 397000, 452391, 513370, 580316, 653621, 733644, 820800, 915517, 1018186, 1129258, 1249197

Offset: 1

Wesley Ivan Hurt and Antonio Osorio, Apr 15 2014

			For a(n) add the numbers 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 + 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               10             46             141        ..   a(n)

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).

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

I:=[1,10,46,141,334,680,1247,2106,3348]; [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[-(5*x^6 + 17*x^5 + 25*x^4 + 30*x^3 + 19*x^2 + 7*x + 1)/((x - 1)^5*(x^2 + x + 1)^2), {x, 0, 50}], x] (* Wesley Ivan Hurt, Jun 13 2014 *)
LinearRecurrence[{3, -3, 3, -6, 6, -3, 3, -3, 1}, {1, 10, 46, 141, 334, 680, 1247, 2106, 3348}, 50] (* Vincenzo Librandi, Aug 29 2015 *)
Table[Total[IntegerPartitions[4 n,{4}][[;;,2]]],{n,40}] (* Harvey P. Dale, Aug 17 2024 *)

Vec(-x*(5*x^6+17*x^5+25*x^4+30*x^3+19*x^2+7*x+1)/((x-1)^5*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Apr 16 2014

G.f.: -x*(5*x^6+17*x^5+25*x^4+30*x^3+19*x^2+7*x+1) / ((x-1)^5*(x^2+x+1)^2). - Colin Barker, Apr 16 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)) for n>1. Then a(1) = 1, with a(n) = a(n-1) + b(n-1)/(4n-4) + Sum_{j=0..2n} (Sum_{i=j+1..floor((4n-2-j)/2)} i * (floor((sign((floor((4n-2-j)/2)-j))+ 2)/2)) ), for n>1. - Wesley Ivan Hurt, Jun 27 2014

A239195 Sum of the next to smallest parts in the partitions of 4n into 4 parts with smallest part = 1.

Original entry on oeis.org

1, 5, 17, 42, 78, 134, 215, 315, 447, 616, 812, 1052, 1341, 1665, 2045, 2486, 2970, 3522, 4147, 4823, 5579, 6420, 7320, 8312, 9401, 10557, 11817, 13186, 14630, 16190, 17871, 19635, 21527, 23552, 25668, 27924, 30325, 32825, 35477, 38286, 41202, 44282, 47531

Offset: 1

Wesley Ivan Hurt, Mar 11 2014

			For a(n) add the numbers in the third 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               5              17              42        ..   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, A239057, A239059, A239186.

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)) - ((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i)) - ((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i) + (i + 2) (Floor[(4 n - 2 - i)/2] - i))/(4 n)) (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},{1,5,17,42,78,134,215,315},60] (* Harvey P. Dale, Jul 05 2025 *)

Vec(x*(4*x^5+5*x^4+11*x^3+8*x^2+3*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Sep 22 2014

G.f.: x*(4*x^5+5*x^4+11*x^3+8*x^2+3*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Mar 12 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, Jul 08 2025

A240711 Sum of the largest parts in the partitions of 4n into 4 parts with smallest part = 1.

Original entry on oeis.org

1, 15, 62, 163, 333, 596, 973, 1475, 2130, 2959, 3969, 5192, 6649, 8343, 10310, 12571, 15125, 18012, 21253, 24843, 28826, 33223, 38025, 43280, 49009, 55199, 61902, 69139, 76893, 85220, 94141, 103635, 113762, 124543, 135953, 148056, 160873, 174375, 188630

Offset: 1

Wesley Ivan Hurt, Apr 10 2014

			For a(n) add the parts in the first 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               15             62             163       ..   a(n)

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, A239059, A240707.

b[n_] := Sum[((4 n - 2 - i)*Floor[(4 n - 2 - i)/2] - i (4 n - 2 - i)) (Floor[(Sign[(Floor[(4 n - 2 - i)/2] - i)] + 2)/2]), {i, 0, 2 n}]; c[1] = 1; 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[b[n] - c[n], {n, 50}]

Vec(x*(7*x^6+27*x^5+43*x^4+52*x^3+33*x^2+13*x+1)/((x-1)^4*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Apr 11 2014

G.f.: x*(7*x^6+27*x^5+43*x^4+52*x^3+33*x^2+13*x+1) / ((x-1)^4*(x^2+x+1)^2). - Colin Barker, Apr 11 2014

cp's OEIS Frontend

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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A239186 Sum of the largest two parts in the partitions of 4n into 4 parts with smallest part equal to 1.

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A239667 Sum of the largest parts of the partitions of 4n into 4 parts.

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

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A241084 Sum of the second largest parts of the partitions of 4n into 4 parts.

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A239195 Sum of the next to smallest parts in the partitions of 4n into 4 parts with smallest part = 1.

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