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.

Previous Showing 11-20 of 23 results. Next

A308989 Sum of all the parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 8, 9, 20, 33, 60, 91, 154, 225, 352, 493, 720, 988, 1400, 1869, 2552, 3358, 4464, 5750, 7488, 9504, 12152, 15225, 19140, 23684, 29408, 35970, 44098, 53445, 64836, 77848, 93556, 111423, 132760, 156948, 185514, 217838, 255728, 298350
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A026814, A308990, A308991, A308992, A308994, A308995, A308996, A308997, A308998.

Programs

  • Mathematica
    Table[Total[Flatten[IntegerPartitions[n,{8}]]],{n,0,50}] (* Harvey P. Dale, Jan 12 2022 *)

Formula

a(n) = n * Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} 1.
a(n) = n * A026814(n).
a(n) = A308990(n) + A308991(n) + A308992(n) + A308994(n) + A308995(n) + A308996(n) + A308997(n) + A308998(n).

A060027 Expansion of (1-x-x^N)/((1-x)(1-x^2)(1-x^3)...(1-x^N)) for N = 8.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 13, 12, 16, 15, 18, 15, 18, 12, 12, 2, -3, -20, -31, -59, -81, -122, -160, -222, -280, -369, -457, -581, -708, -878, -1055, -1286, -1528, -1833, -2158, -2559, -2985, -3504, -4059, -4721, -5433, -6271, -7172, -8224, -9355, -10660, -12067
Offset: 0

Views

Author

N. J. A. Sloane, Mar 17 2001

Keywords

Comments

Difference of the number of partitions of n+7 into 7 parts and the number of partitions of n+7 into 8 parts. - Wesley Ivan Hurt, Apr 16 2019

Links

Crossrefs

Cf. A026813, A026814.
Cf. For other values of N: A060022 (N=3), A060023 (N=4), A060024 (N=5), A060025 (N=6), A060026 (N=7), this sequence (N=8), A060028 (N=9), A060029 (N=10).

Programs

  • Mathematica
    With[{nn=8},CoefficientList[Series[(1-x-x^nn)/Times@@(1-x^Range[nn]),{x,0,60}],x]] (* Harvey P. Dale, May 15 2016 *)

Formula

a(n) = A026813(n+7) - A026814(n+7). - Wesley Ivan Hurt, Apr 16 2019

A308990 Sum of the smallest parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 23, 30, 42, 55, 75, 96, 127, 161, 209, 260, 330, 407, 509, 621, 765, 925, 1128, 1350, 1627, 1934, 2310, 2725, 3227, 3782, 4447, 5178, 6044, 7000, 8122, 9355, 10791, 12370, 14196, 16196, 18494, 21012, 23887
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A026814, A308989, A308991, A308992, A308994, A308995, A308996, A308997, A308998.

Programs

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

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} p.
a(n) = A308989(n) - A308991(n) - A308992(n) - A308994(n) - A308995(n) - A308996(n) - A308997(n) - A308998(n).

A308991 Sum of the seventh largest parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 24, 32, 45, 60, 82, 107, 143, 184, 240, 303, 387, 484, 609, 753, 934, 1142, 1401, 1695, 2056, 2468, 2967, 3532, 4208, 4974, 5882, 6904, 8105, 9458, 11033, 12798, 14840, 17124, 19750, 22674, 26018, 29735
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A026814, A308989, A308990, A308992, A308994, A308995, A308996, A308997, A308998.

Programs

  • Mathematica
    Table[Total[IntegerPartitions[n,{8}][[All,7]]],{n,0,60}] (* Harvey P. Dale, Apr 14 2022 *)

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} o.
a(n) = A308989(n) - A308990(n) - A308992(n) - A308994(n) - A308995(n) - A308996(n) - A308997(n) - A308998(n).

A308992 Sum of the sixth largest parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 17, 26, 35, 50, 67, 94, 123, 167, 216, 285, 362, 469, 589, 749, 931, 1165, 1431, 1771, 2152, 2630, 3171, 3836, 4585, 5497, 6521, 7753, 9134, 10775, 12615, 14784, 17202, 20030, 23182, 26837, 30897, 35581, 40769
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A026814, A308989, A308990, A308991, A308994, A308995, A308996, A308997, A308998.

Programs

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

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} m.
a(n) = A308989(n) - A308990(n) - A308991(n) - A308994(n) - A308995(n) - A308996(n) - A308997(n) - A308998(n).

A308994 Sum of the fifth largest parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 13, 19, 29, 40, 58, 79, 111, 148, 201, 264, 349, 449, 583, 739, 943, 1181, 1482, 1833, 2273, 2780, 3405, 4126, 5002, 6006, 7215, 8593, 10235, 12101, 14300, 16795, 19713, 23003, 26825, 31124, 36083, 41638, 48012
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A026814, A308989, A308990, A308991, A308992, A308995, A308996, A308997, A308998.

Programs

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

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} l.
a(n) = A308989(n) - A308990(n) - A308991(n) - A308992(n) - A308995(n) - A308996(n) - A308997(n) - A308998(n).

A308995 Sum of the fourth largest parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 9, 15, 22, 35, 48, 71, 97, 139, 185, 254, 334, 447, 575, 752, 955, 1227, 1537, 1939, 2401, 2991, 3661, 4500, 5458, 6639, 7977, 9607, 11452, 13673, 16176, 19154, 22511, 26470, 30906, 36096, 41906, 48652, 56171, 64847
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A026814, A308989, A308990, A308991, A308992, A308994, A308996, A308997, A308998.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k, {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]
Table[Total[IntegerPartitions[n,{8}][[;;,4]]],{n,0,60}] (* Harvey P. Dale, Nov 20 2024 *)

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} k.
a(n) = A308989(n) - A308990(n) - A308991(n) - A308992(n) - A308994(n) - A308996(n) - A308997(n) - A308998(n).

A308996 Sum of the third largest parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 11, 19, 28, 44, 63, 92, 128, 183, 246, 337, 448, 597, 774, 1012, 1291, 1656, 2085, 2627, 3264, 4064, 4987, 6127, 7450, 9055, 10901, 13126, 15669, 18701, 22157, 26228, 30858, 36279, 42397, 49509, 57527, 66773, 77148
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A026814, A308989, A308990, A308991, A308992, A308994, A308995, A308997, A308998.

Programs

  • Mathematica
    Table[Total[IntegerPartitions[n,{8}][[All,3]]],{n,0,50}] (* Harvey P. Dale, Jul 19 2020 *)

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} j.
a(n) = A308989(n) - A308990(n) - A308991(n) - A308992(n) - A308994(n) - A308995(n) - A308997(n) - A308998(n).

A308997 Sum of the second largest parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 15, 27, 39, 63, 89, 133, 183, 264, 353, 488, 644, 864, 1116, 1465, 1863, 2397, 3009, 3802, 4713, 5877, 7200, 8859, 10753, 13084, 15731, 18956, 22603, 26993, 31948, 37839, 44477, 52307, 61082, 71349, 82842, 96177
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A026814, A308989, A308990, A308991, A308992, A308994, A308995, A308996, A308998.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[i, {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]
Table[Total[IntegerPartitions[n,{8}][[;;,2]]],{n,0,50}] (* Harvey P. Dale, Jun 22 2024 *)

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} i.
a(n) = A308989(n) - A308990(n) - A308991(n) - A308992(n) - A308994(n) - A308995(n) - A308996(n) - A308998(n).

A308998 Sum of the largest parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 9, 17, 27, 46, 69, 108, 156, 229, 319, 452, 611, 835, 1107, 1473, 1911, 2490, 3176, 4062, 5108, 6426, 7975, 9903, 12145, 14894, 18085, 21943, 26391, 31720, 37829, 45076, 53350, 63069, 74124, 87020, 101607, 118504, 137561
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

Links

Crossrefs

Cf. A026814, A308989, A308990, A308991, A308992, A308994, A308995, A308996, A308997.

Programs

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

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} (n-i-j-k-l-m-o-p).
a(n) = A308989(n) - A308990(n) - A308991(n) - A308992(n) - A308994(n) - A308995(n) - A308996(n) - A308997(n).
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