A235988 -id:A235988 - 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

A236364 Sum of all the middle parts in the partitions of 3n into 3 parts.

Original entry on oeis.org

1, 5, 18, 40, 80, 135, 217, 320, 459, 625, 836, 1080, 1378, 1715, 2115, 2560, 3077, 3645, 4294, 5000, 5796, 6655, 7613, 8640, 9775, 10985, 12312, 13720, 15254, 16875, 18631, 20480, 22473, 24565, 26810, 29160, 31672, 34295, 37089, 40000, 43091, 46305, 49708

Offset: 1

Wesley Ivan Hurt, Jan 23 2014

			Add second columns for a(n):
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
------------------------------------------------------------------------
    1           5          18           40          80      ..   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,-4,1,2,-1).

Cf. A000330, A019298, A235988.

A236364:=n->n*(n+1)*(2*n+1)/6 - floor((n-1)/2) * (4*floor((n-1)/2)^2 + (3*n+6)*floor((n-1)/2) - 6*n^2 + 3*n + 2)/6; seq(A236364(n), n=1..100);

Table[Sum[i^2, {i, n}] + Sum[(n + i) (n - 2 i), {i, Floor[(n - 1)/2]}], {n, 100}]
CoefficientList[Series[(x^4 + 3 x^3 + 7 x^2 + 3 x + 1)/ ((x - 1)^4 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 18 2014 *)

Vec(x*(x^4+3*x^3+7*x^2+3*x+1)/((x-1)^4*(x+1)^2) + O(x^100)) \\ Colin Barker, Jan 23 2014

a(n) = A000330(n) + Sum_{i=1..floor((n-1)/2)} (n + i)*(n - 2i).
a(n) = n*(n+1)*(2*n+1)/6 - floor((n-1)/2) * (4*floor((n-1)/2)^2 + 3*(n+2)*floor((n-1)/2) - 6*n^2 + 3*n + 2)/6.
G.f.: x*(x^4+3*x^3+7*x^2+3*x+1)/((x-1)^4*(x+1)^2). - Joerg Arndt, Jan 23 2014
a(n) = (n*(3-3*(-1)^n+10*n^2))/16. - Colin Barker, Jan 23 2014
a(n) = (n^3 + ceiling(n/2)^3 + floor(n/2)^3)/2. - Wesley Ivan Hurt, Apr 15 2016
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Wesley Ivan Hurt, Nov 19 2021

Name clarified by Wesley Ivan Hurt, Apr 16 2016

A236758 Number of partitions of 3*n into 3 parts with smallest part prime.

Original entry on oeis.org

0, 1, 3, 6, 10, 14, 20, 25, 32, 37, 45, 51, 61, 68, 79, 86, 98, 106, 120, 129, 144, 153, 169, 179, 196, 206, 223, 233, 251, 262, 282, 294, 315, 327, 348, 360, 382, 395, 418, 431, 455, 469, 495, 510, 537, 552, 580, 596, 625, 641, 670, 686, 716, 733, 764, 781

Offset: 1

Wesley Ivan Hurt, Jan 30 2014

			Count the primes in last column for a(n):
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
---------------------------------------------------------------------
    0           1           3           6           10      ..   a(n)

Index entries for sequences related to partitions

Cf. A019298, A235988, A236364, A236762, A010051 (for function isprime).

with(numtheory); A236758:=n->sum((pi(n) - pi(n-1)) * (2*n - 2*i + 1 - floor((n - i + 1)/2)), i=1..n); seq(A236758(n), n=1..100);

Table[Sum[(PrimePi[i] - PrimePi[i - 1]) (2 n - 2 i + 1 - Floor[(n - i + 1)/2]), {i, n}], {n, 100}]

def a(n): return sum(1 for L in Partitions(3*n,length=3).list() if is_prime(L[2]))

a(n) = Sum_{i=1..n} A010051(i) * (2*n - 2*i + 1 - floor((n - i + 1)/2)).

A236762 Number of partitions of 3n into 3 parts with the middle part prime.

Original entry on oeis.org

0, 2, 5, 7, 11, 14, 17, 19, 23, 29, 35, 40, 47, 53, 59, 67, 76, 82, 88, 93, 100, 109, 118, 124, 131, 140, 149, 160, 173, 185, 197, 208, 220, 232, 244, 258, 273, 285, 297, 311, 327, 342, 357, 369, 382, 397, 412, 426, 442, 460, 478, 496, 515, 533, 551, 571

Offset: 1

Wesley Ivan Hurt, Jan 30 2014

			Count the primes in the second columns for a(n):
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
--------------------------------------------------------------------
    0           2           5           7          11      ..   a(n)

Index entries for sequences related to partitions

Cf. A019298, A235988, A236364, A236758, A010051.

with(numtheory); A236762:=n->sum( i * (pi(i) - pi(i - 1)), i = 1..n) +
sum( (pi(n + i) - pi(n + i - 1)) * (n - 2*i), i = 1..floor((n - 1)/2) ); seq(A236762(n), n=1..100);

Table[Sum[i (PrimePi[i] - PrimePi[i - 1]), {i, n}] + Sum[(PrimePi[n + i] - PrimePi[n + i - 1]) (n - 2 i), {i, Floor[(n - 1)/2]}], {n, 100}]

def a(n): return sum(1 for L in Partitions(3*n,length=3).list() if is_prime(L[1])) # Ralf Stephan, Feb 03 2014

a(n) = Sum_{i=1..n} i * A010051(i) + Sum_{i=1..floor((n - 1)/2)} A010051(n + i) * (n - 2i).

A256235 Sum of all the parts in the partitions of 5n into 5 parts.

Original entry on oeis.org

0, 5, 70, 450, 1680, 4800, 11310, 23590, 44600, 78615, 130550, 207075, 315600, 465790, 667940, 935250, 1281520, 1723970, 2280330, 2972455, 3822500, 4857510, 6104560, 7596325, 9365400, 11450750, 13890760, 16731225, 20017060, 23801315, 28135800, 33081495

Offset: 0

Colin Barker, Mar 20 2015

			For n=2 there are 7 partitions of 5*2 = 10, so a(2) = 7*10 = 70.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,1,0,-4,-1,-1,4,4,-1,-1,-4,0,1,1,1,-1).

Cf. A235988, A238328, A256225, A256239.

Plus @@ Total /@ IntegerPartitions[5 #, {5}] & /@ Range[0, 31] (* Michael De Vlieger, Mar 20 2015 *)
CoefficientList[Series[5 x (2 x^14 + 19 x^13 + 97 x^12 + 277 x^11 + 591 x^10 + 955 x^9 + 1267 x^8 + 1355 x^7 + 1217 x^6 + 880 x^5 + 520 x^4 + 231 x^3 + 75 x^2 + 13 x + 1) / ((x - 1)^6 (x + 1)^3 (x^2 + 1)^2 (x^2 + x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{1,1,1,0,-4,-1,-1,4,4,-1,-1,-4,0,1,1,1,-1},{0,5,70,450,1680,4800,11310,23590,44600,78615,130550,207075,315600,465790,667940,935250,1281520},40] (* Harvey P. Dale, Jun 14 2016 *)

concat(0, Vec(5*x*(2*x^14 +19*x^13 +97*x^12 +277*x^11 +591*x^10 +955*x^9 +1267*x^8 +1355*x^7 +1217*x^6 +880*x^5 +520*x^4 +231*x^3 +75*x^2 +13*x +1) / ((x -1)^6*(x +1)^3*(x^2 +1)^2*(x^2 +x +1)^2) + O(x^100)))

a(n) = 5*n*A256225(n).

G.f.: 5*x*(2*x^14 +19*x^13 +97*x^12 +277*x^11 +591*x^10 +955*x^9 +1267*x^8 +1355*x^7 +1217*x^6 +880*x^5 +520*x^4 +231*x^3 +75*x^2 +13*x +1) / ((x -1)^6*(x +1)^3*(x^2 +1)^2*(x^2 +x +1)^2).

A236370 Sum of the largest parts in the partitions of 3n into 3 parts.

Original entry on oeis.org

1, 9, 34, 81, 163, 282, 454, 678, 973, 1335, 1786, 2319, 2959, 3696, 4558, 5532, 6649, 7893, 9298, 10845, 12571, 14454, 16534, 18786, 21253, 23907, 26794, 29883, 33223, 36780, 40606, 44664, 49009, 53601, 58498, 63657, 69139, 74898, 80998, 87390, 94141

Offset: 1

Wesley Ivan Hurt, Jan 23 2014

			Add first columns for a(n)..
                                               13 + 1 + 1
                                               12 + 2 + 1
                                               11 + 3 + 1
                                               10 + 4 + 1
                                                9 + 5 + 1
                                                8 + 6 + 1
                                                7 + 7 + 1
                                   10 + 1 + 1  11 + 2 + 2
                                    9 + 2 + 1  10 + 3 + 2
                                    8 + 3 + 1   9 + 4 + 2
                                    7 + 4 + 1   8 + 5 + 2
                                    6 + 5 + 1   7 + 6 + 2
                        7 + 1 + 1   8 + 2 + 2   9 + 3 + 3
                        6 + 2 + 1   7 + 3 + 2   8 + 4 + 3
                        5 + 3 + 1   6 + 4 + 2   7 + 5 + 3
                        4 + 4 + 1   5 + 5 + 2   6 + 6 + 3
            4 + 1 + 1   5 + 2 + 2   6 + 3 + 3   7 + 4 + 4
            3 + 2 + 1   4 + 3 + 2   5 + 4 + 3   6 + 5 + 4
1 + 1 + 1   2 + 2 + 2   3 + 3 + 3   4 + 4 + 4   5 + 5 + 5
   3(1)        3(2)        3(3)        3(4)        3(5)     ..   3n
---------------------------------------------------------------------
    1           9          34           81          163      ..  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,-4,1,2,-1).

Cf. A019298, A235988, A236364, A236758, A236762, A237264.

Table[3 n (n^2 - Floor[n^2/4]) - Sum[2 i^2 - Floor[i^2/4], {i, n}] -
  Sum[(n + i) (n - 2 i), {i, Floor[(n - 1)/2]}], {n, 100}]
LinearRecurrence[{2,1,-4,1,2,-1},{1,9,34,81,163,282},50] (* Harvey P. Dale, Nov 11 2017 *)

Vec(x*(2*x^4+8*x^3+15*x^2+7*x+1)/((x-1)^4*(x+1)^2) + O(x^100)) \\ Colin Barker, Jan 24 2014

a(n) = 3n * (n^2 - floor(n^2/4)) - Sum_{i=1..n} (2*i^2 - floor(i^2/4)) - Sum_{i=1..floor((n-1)/2)} (n + i) * (n - 2i).
From Colin Barker, Jan 24 2014: (Start)
a(n) = (-1+(-1)^n-(1+3*(-1)^n)*n-6*n^2+22*n^3)/16.
G.f.: x*(2*x^4+8*x^3+15*x^2+7*x+1) / ((x-1)^4*(x+1)^2). (End)
a(n) = Sum_{j=0..n-2} (Sum_{i=n+1+floor(j/2)-floor(1/j+1)..n+2*(j+1)} i), n > 1. - Wesley Ivan Hurt, Feb 10 2014
a(n) = 2*a(n-1)+a(n-2)-4*a(n-3)+a(n-4)+2*a(n-5)-a(n-6). - Wesley Ivan Hurt, Nov 19 2021

A256239 Sum of all the parts in the partitions of 6n into 6 parts.

Original entry on oeis.org

0, 6, 132, 1044, 4776, 15960, 43416, 102144, 215712, 419040, 761520, 1310628, 2155752, 3412656, 5228076, 7784910, 11307648, 16068264, 22392504, 30666570, 41344080, 54953640, 72106452, 93504798, 119950416, 152353650, 191742720, 239273514, 296239776, 364083690

Offset: 0

Colin Barker, Mar 20 2015

			For n=2 there are 11 partitions of 6*2 = 12, so a(2) = 11*12 = 132.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,3,-9,3,10,-10,-3,9,-3,-5,5,1,-3,1).

Cf. A235988, A238328, A256225, A256235.

Plus @@ Total /@ IntegerPartitions[6 #, {6}] & /@ Range[0, 29] (* Michael De Vlieger, Mar 20 2015 *)
CoefficientList[Series[- 6 x (9 x^13 + 77 x^12 + 247 x^11 + 485 x^10 + 744 x^9 + 990 x^8 + 1109 x^7 + 1029 x^6 + 809 x^5 + 551 x^4 + 301 x^3 + 109 x^2 + 19 x + 1) / ((x - 1)^7 (x + 1)^2 (x^4 + x^3 + x^2 + x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 20 2015 *)
LinearRecurrence[{3,-1,-5,5,3,-9,3,10,-10,-3,9,-3,-5,5,1,-3,1},{0,6,132,1044,4776,15960,43416,102144,215712,419040,761520,1310628,2155752,3412656,5228076,7784910,11307648},30] (* Harvey P. Dale, Mar 07 2025 *)

concat(0, Vec(-6*x*(9*x^13 +77*x^12 +247*x^11 +485*x^10 +744*x^9 +990*x^8 +1109*x^7 +1029*x^6 +809*x^5 +551*x^4 +301*x^3 +109*x^2 +19*x +1) / ((x -1)^7*(x +1)^2*(x^4 +x^3 +x^2 +x +1)^2) + O(x^100)))

a(n) = 6*n*A256226(n).

G.f.: -6*x*(9*x^13 +77*x^12 +247*x^11 +485*x^10 +744*x^9 +990*x^8 +1109*x^7 +1029*x^6 +809*x^5 +551*x^4 +301*x^3 +109*x^2 +19*x +1) / ((x -1)^7*(x +1)^2*(x^4 +x^3 +x^2 +x +1)^2).

A256287 Number of partitions of 7n into 7 parts.

Original entry on oeis.org

0, 1, 15, 105, 436, 1367, 3539, 8033, 16475, 31275, 55748, 94425, 153192, 239691, 363446, 536375, 772909, 1090592, 1510201, 2056462, 2758123, 3648814, 4767088, 6157387, 7870067, 9962502, 12499033, 15552247, 19202869, 23541165, 28666799, 34690401, 41733315

Offset: 0

Colin Barker, Mar 21 2015

			For n=2, the 15 partitions of 14 are [1,1,1,1,1,1,8], [1,1,1,1,1,2,7], ..., [1,2,2,2,2,2,3], [2,2,2,2,2,2,2].

Ray Chandler, Table of n, a(n) for n = 0..50
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -2, 2, 1, 0, 0, 0, -1, -2, 2, -1, 1, 0, 1, 0, -2, 1).

Cf. A235988, A238340, A256225, A256226, A256288.

Length /@ (Total /@ IntegerPartitions[7 #, {7}] & /@ Range[0, 24]) (* Michael De Vlieger, Mar 21 2015 *)

concat(0, vector(35, n, k=0; forpart(p=7*n, k++, , [7,7]); k))

A256288 Sum of all the parts in the partitions of 7n into 7 parts.

Original entry on oeis.org

0, 7, 210, 2205, 12208, 47845, 148638, 393617, 922600, 1970325, 3902360, 7270725, 12868128, 21811881, 35617708, 56319375, 86565808, 129780448, 190285326, 273509446, 386137220, 536375658, 734131552, 991339307, 1322171256, 1743437850, 2274824006, 2939374683

Offset: 0

Colin Barker, Mar 21 2015

nonn

			For n=2 there are 15 partitions of 7*2 = 14, so a(2) = 15*14 = 210.

Ray Chandler, Table of n, a(n) for n = 0..50

Cf. A235988, A238328, A256235, A256287.

Plus @@ Total /@ IntegerPartitions[7 #, {7}] & /@ Range[0, 24] (* Michael De Vlieger, Mar 21 2015 *)

concat(0, vector(35, n, k=0; forpart(p=7*n, k++, , [7,7]); 7*n*k))

a(n) = 7*n*A256287(n).

cp's OEIS Frontend

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

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Magma

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A236364 Sum of all the middle parts in the partitions of 3n into 3 parts.

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Maple

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A236758 Number of partitions of 3*n into 3 parts with smallest part prime.

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Sage

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A236762 Number of partitions of 3n into 3 parts with the middle part prime.

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Maple

Mathematica

Sage

Formula

A256235 Sum of all the parts in the partitions of 5n into 5 parts.

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Mathematica

PARI

Formula

A236370 Sum of the largest parts in the partitions of 3n into 3 parts.

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Mathematica

PARI

Formula

A256239 Sum of all the parts in the partitions of 6n into 6 parts.

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