cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A326588 Sum of all the parts in the partitions of n into 10 parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

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

Programs

  • 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}]

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).

A326589 Sum of the smallest parts of the partitions of n into 10 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 56, 77, 100, 133, 171, 223, 282, 362, 453, 573, 709, 884, 1084, 1337, 1626, 1984, 2394, 2896, 3468, 4163, 4951, 5897, 6972, 8249, 9696, 11402, 13330, 15586, 18131, 21091, 24417, 28264, 32580
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

Cf. A026816, A326588, A326590, A326591, A326592, A326593, A326594, A326595, A326596, A326597, A326598.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[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}]

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)} r.
a(n) = A326588(n) - A326590(n) - A326591(n) - A326592(n) - A326593(n) - A326594(n) - A326595(n) - A326596(n) - A326597(n) - A326598(n).

A326590 Sum of the ninth largest parts of the partitions of n into 10 parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

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

Programs

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

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).

A326591 Sum of the eighth largest parts of the partitions of n into 10 parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

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

Programs

  • 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}]

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).

A326592 Sum of the seventh largest parts in the partitions of n into 10 parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

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

Programs

  • 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}]
Table[Total[IntegerPartitions[n,{10}][[;;,7]]],{n,0,60}] (* Harvey P. Dale, Jul 14 2025 *)

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).

A326593 Sum of the sixth largest parts of the partitions of n into 10 parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

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

Programs

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

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).

A326594 Sum of the fifth largest parts of the partitions of n into 10 parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

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

Programs

  • 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}]

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).

A326595 Sum of the fourth largest parts of the partitions of n into 10 parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

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

Programs

  • 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}]

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).

A326596 Sum of the third largest parts of the partitions of n into 10 parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

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

Programs

  • 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}]
Table[Total[IntegerPartitions[n,{10}][[;;,3]]],{n,0,50}] (* Harvey P. Dale, Jul 23 2025 *)

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).

A326598 Sum of the largest parts of the partitions of n into 10 parts.

Original entry on oeis.org

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
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 13 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    Table[Total[IntegerPartitions[n,{10}][[;;,1]]],{n,0,50}] (* Harvey P. Dale, May 02 2025 *)

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).
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