A326589 -id:A326589

A326588

Sum of all the parts in the partitions of n into 10 parts.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 11, 24, 39, 70, 105, 176, 255, 396, 570, 840, 1155, 1650, 2231, 3072, 4100, 5512, 7209, 9520, 12267, 15900, 20243, 25824, 32472, 40936, 50925, 63396, 78144, 96292, 117585, 143600, 173922, 210546, 253184, 304128, 363150

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,11

LINKS

Table of n, a(n) for n=0..45.

Index entries for sequences related to partitions

FORMULA

a(n) = n * Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} 1.

a(n) = n * A026816(n).

a(n) = A326589(n) + A326590(n) + A326591(n) + A326592(n) + A326593(n) + A326594(n) + A326595(n) + A326596(n) + A326597(n) + A326598(n).

MATHEMATICA

Table[n * Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[1, {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

CROSSREFS

Cf. A026816, A326589, A326590, A326591, A326592, A326593, A326594, A326595, A326596, A326597, A326598.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 13 2019

STATUS

approved

A326590

Sum of the ninth largest parts of the partitions of n into 10 parts.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 31, 44, 58, 80, 105, 140, 182, 238, 304, 393, 496, 630, 787, 986, 1219, 1512, 1853, 2273, 2765, 3362, 4055, 4894, 5860, 7016, 8351, 9931, 11746, 13885, 16330, 19188, 22452, 26242, 30549, 35531

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,13

LINKS

Table of n, a(n) for n=0..52.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} q.

a(n) = A326588(n) - A326589(n) - A326591(n) - A326592(n) - A326593(n) - A326594(n) - A326595(n) - A326596(n) - A326597(n) - A326598(n).

MATHEMATICA

Table[Total[IntegerPartitions[n, {10}][[;; , 9]]], {n, 0, 60}] (* Harvey P. Dale, Mar 18 2023 *)

CROSSREFS

Cf. A026816, A326588, A326589, A326591, A326592, A326593, A326594, A326595, A326596, A326597, A326598.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 13 2019

STATUS

approved

A326591

Sum of the eighth largest parts of the partitions of n into 10 parts.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 23, 32, 46, 61, 85, 112, 151, 197, 261, 335, 437, 554, 710, 891, 1125, 1398, 1747, 2151, 2657, 3246, 3972, 4812, 5840, 7023, 8455, 10104, 12076, 14339, 17029, 20102, 23724, 27857, 32694, 38190, 44588

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,13

LINKS

Table of n, a(n) for n=0..52.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} p.

a(n) = A326588(n) - A326589(n) - A326590(n) - A326592(n) - A326593(n) - A326594(n) - A326595(n) - A326596(n) - A326597(n) - A326598(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[p, {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

CROSSREFS

Cf. A026816, A326588, A326589, A326590, A326592, A326593, A326594, A326595, A326596, A326597, A326598.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 13 2019

STATUS

approved

A326592

Sum of the seventh largest parts in the partitions of n into 10 parts.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 24, 34, 49, 66, 92, 123, 167, 220, 293, 380, 497, 636, 818, 1035, 1312, 1642, 2059, 2551, 3162, 3884, 4769, 5806, 7068, 8539, 10310, 12370, 14826, 17670, 21038, 24920, 29482, 34725, 40848, 47852, 55989

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,13

LINKS

Table of n, a(n) for n=0..52.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} o.

a(n) = A326588(n) - A326589(n) - A326590(n) - A326591(n) - A326593(n) - A326594(n) - A326595(n) - A326596(n) - A326597(n) - A326598(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[o, {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

CROSSREFS

Cf. A026816, A326588, A326589, A326590, A326591, A326593, A326594, A326595, A326596, A326597, A326598.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 13 2019

STATUS

approved

A326593

Sum of the sixth largest parts of the partitions of n into 10 parts.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 17, 26, 37, 54, 73, 104, 139, 191, 253, 340, 442, 584, 749, 970, 1232, 1571, 1971, 2486, 3087, 3844, 4734, 5835, 7119, 8699, 10530, 12753, 15332, 18426, 21998, 26259, 31153, 36938, 43575, 51360, 60250

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,13

LINKS

Table of n, a(n) for n=0..51.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} m.

a(n) = A326588(n) - A326589(n) - A326590(n) - A326591(n) - A326592(n) - A326594(n) - A326595(n) - A326596(n) - A326597(n) - A326598(n).

MATHEMATICA

Table[Total[IntegerPartitions[n, {10}][[All, 6]]], {n, 0, 60}] (* Harvey P. Dale, Dec 20 2020 *)

CROSSREFS

Cf. A026816, A326588, A326589, A326590, A326591, A326592, A326594, A326595, A326596, A326597, A326598.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 13 2019

STATUS

approved

A326594

Sum of the fifth largest parts of the partitions of n into 10 parts.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 13, 19, 29, 42, 62, 85, 121, 164, 226, 303, 407, 534, 706, 912, 1184, 1511, 1930, 2433, 3072, 3831, 4776, 5900, 7281, 8909, 10898, 13223, 16031, 19312, 23231, 27787, 33194, 39444, 46806, 55292, 65219, 76603

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,13

LINKS

Table of n, a(n) for n=0..51.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} l.

a(n) = A326588(n) - A326589(n) - A326590(n) - A326591(n) - A326592(n) - A326593(n) - A326595(n) - A326596(n) - A326597(n) - A326598(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[l, {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

CROSSREFS

Cf. A026816, A326588, A326589, A326590, A326591, A326592, A326593, A326595, A326596, A326597, A326598.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 13 2019

STATUS

approved

A326595

Sum of the fourth largest parts of the partitions of n into 10 parts.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 9, 15, 22, 35, 50, 75, 103, 149, 202, 281, 376, 510, 669, 889, 1149, 1499, 1913, 2453, 3093, 3917, 4886, 6106, 7544, 9330, 11419, 13989, 16979, 20614, 24837, 29912, 35785, 42790, 50857, 60399, 71360, 84233, 98952

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,13

LINKS

Table of n, a(n) for n=0..51.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} k.

a(n) = A326588(n) - A326589(n) - A326590(n) - A326591(n) - A326592(n) - A326593(n) - A326594(n) - A326596(n) - A326597(n) - A326598(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k, {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

CROSSREFS

Cf. A026816, A326588, A326589, A326590, A326591, A326592, A326593, A326594, A326596, A326597, A326598.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 13 2019

STATUS

approved

A326596

Sum of the third largest parts of the partitions of n into 10 parts.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 11, 19, 28, 44, 65, 96, 134, 194, 265, 367, 496, 670, 883, 1173, 1521, 1980, 2537, 3248, 4104, 5194, 6488, 8101, 10025, 12387, 15175, 18582, 22570, 27385, 33020, 39745, 47569, 56861, 67602, 80253, 94849, 111914

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,13

LINKS

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

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} j.

a(n) = A326588(n) - A326589(n) - A326590(n) - A326591(n) - A326592(n) - A326593(n) - A326594(n) - A326595(n) - A326597(n) - A326598(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[j, {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

CROSSREFS

Cf. A026816, A326588, A326589, A326590, A326591, A326592, A326593, A326594, A326595, A326597, A326598.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 13 2019

STATUS

approved

A326597

Sum of the second largest parts of the partitions of n into 10 parts.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 15, 27, 39, 63, 91, 137, 190, 277, 376, 525, 704, 956, 1255, 1671, 2160, 2818, 3599, 4616, 5819, 7369, 9187, 11480, 14179, 17527, 21441, 26256, 31851, 38649, 46543, 56022, 66980, 80050, 95083, 112860, 133266

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,13

LINKS

Table of n, a(n) for n=0..49.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} i.

a(n) = A326588(n) - A326589(n) - A326590(n) - A326591(n) - A326592(n) - A326593(n) - A326594(n) - A326595(n) - A326596(n) - A326598(n).

MATHEMATICA

Table[Total[IntegerPartitions[n, {10}][[;; , 2]]], {n, 0, 50}] (* Harvey P. Dale, Nov 19 2023 *)

CROSSREFS

Cf. A026816, A326588, A326589, A326590, A326591, A326592, A326593, A326594, A326595, A326596, A326598.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 13 2019

STATUS

approved

A326598

Sum of the largest parts of the partitions of n into 10 parts.

+10
10

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 9, 17, 27, 46, 69, 108, 158, 234, 329, 471, 645, 891, 1198, 1614, 2125, 2808, 3637, 4718, 6029, 7699, 9709, 12243, 15265, 19013, 23473, 28933, 35381, 43211, 52396, 63436, 76343, 91710, 109580, 130720, 155171, 183884

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,12

LINKS

Table of n, a(n) for n=0..48.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} (n-i-j-k-l-m-o-p-q-r).

a(n) = A326588(n) - A326589(n) - A326590(n) - A326591(n) - A326592(n) - A326593(n) - A326594(n) - A326595(n) - A326596(n) - A326597(n).

MATHEMATICA

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(n - i - j - k - l - m - o - p - q - r), {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]

CROSSREFS

Cf. A026816, A326588, A326589, A326590, A326591, A326592, A326593, A326594, A326595, A326596, A326597.

KEYWORD

nonn

AUTHOR

Wesley Ivan Hurt, Jul 13 2019

STATUS

approved