A026813 -id:A026813 - cp's OEIS frontend

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

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

N. J. A. Sloane, Mar 17 2001

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

Ray Chandler, Table of n, a(n) for n = 0..1000
P. A. MacMahon, Perpetual reciprocants, Proc. London Math. Soc., 17 (1886), 139-151; Coll. Papers II, pp. 584-596.
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1, 1, 0, 0, -1, 0, -1, 0, -1, 0, 1, 2, 1, 0, 1, -1, -1, -2, -1, -1, 1, 0, 1, 2, 1, 0, -1, 0, -1, 0, -1, 0, 0, 1, 1, -1).

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).

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

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

A211862 Number of partitions of n into parts <= 7 with the property that all parts have distinct multiplicities.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 12, 14, 19, 25, 26, 39, 46, 51, 65, 84, 87, 116, 123, 147, 171, 216, 220, 281, 306, 364, 402, 496, 511, 636, 678, 793, 861, 1032, 1062, 1273, 1360, 1569, 1683, 1978, 2054, 2428, 2566, 2953, 3118, 3627, 3812, 4378, 4631

Offset: 0

Matthew C. Russell, Apr 25 2012

nonn

			For n=3 the a(3)=2 partitions are {3} and {1,1,1}. Note that {2,1} does not count, as 1 and 2 appear with the same nonzero multiplicity.

Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity partitions.

Cf. A026813, A098859.

Cf. A105637, A211858, A211859, A211860, A211861, A211863.

a211862 n = p 0 [] [1..7] n where
   p m ms _      0 = if m `elem` ms then 0 else 1
   p    []     _ = 0
   p m ms ks'@(k:ks) x
     | x < k       = 0
     | m == 0      = p 1 ms ks' (x - k) + p 0 ms ks x
     | m `elem` ms = p (m + 1) ms ks' (x - k)
     | otherwise   = p (m + 1) ms ks' (x - k) + p 0 (m : ms) ks x
-- Reinhard Zumkeller, Dec 27 2012

A308926 Sum of all the parts in the partitions of n into 7 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 7, 8, 18, 30, 55, 84, 143, 210, 315, 448, 646, 882, 1235, 1640, 2205, 2882, 3772, 4824, 6200, 7800, 9828, 12208, 15138, 18540, 22723, 27520, 33297, 39950, 47845, 56844, 67488, 79534, 93600, 109520, 127920, 148638, 172473, 199144, 229590

Offset: 0

Wesley Ivan Hurt, Jun 30 2019

nonn

Index entries for sequences related to partitions

Cf. A026813, A308927, A308928, A308929, A308930, A308931, A308932, A308933.

Table[Total[Flatten[IntegerPartitions[n,{7}]]],{n,0,50}] (* Harvey P. Dale, Jul 27 2025 *)

a(n) = n * Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} 1.
a(n) = n * A026813(n).
a(n) = A308927(n) + A308928(n) + A308929(n) + A308930(n) + A308931(n) + A308932(n) + A308933(n).

A308927 Sum of the smallest parts in the partitions of n into 7 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 22, 30, 41, 54, 72, 93, 121, 153, 194, 242, 302, 372, 457, 557, 675, 812, 975, 1162, 1381, 1632, 1924, 2254, 2636, 3068, 3562, 4119, 4752, 5463, 6265, 7162, 8170, 9293, 10549, 11942, 13496, 15211, 17115, 19214

Offset: 0

Wesley Ivan Hurt, Jun 30 2019

nonn

Index entries for sequences related to partitions

Cf. A026813, A308926, A308928, A308929, A308930, A308931, A308932, A308933.

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

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

a(n) = A308926(n) - A308928(n) - A308929(n) - A308930(n) - A308931(n) - A308932(n) - A308933(n).

A308928 Sum of the sixth largest parts in the partitions of n into 7 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 17, 24, 33, 46, 61, 84, 109, 144, 184, 237, 298, 379, 470, 585, 719, 882, 1069, 1300, 1560, 1873, 2230, 2653, 3129, 3694, 4326, 5063, 5892, 6848, 7917, 9147, 10513, 12071, 13804, 15765, 17935, 20389, 23088, 26118

Offset: 0

Wesley Ivan Hurt, Jun 30 2019

nonn

Index entries for sequences related to partitions

Cf. A026813, A308926, A308927, A308929, A308930, A308931, A308932, A308933.

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

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

a(n) = A308926(n) - A308927(n) - A308929(n) - A308930(n) - A308931(n) - A308932(n) - A308933(n).

A308929 Sum of the fifth largest parts in the partitions of n into 7 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 13, 19, 27, 38, 54, 73, 101, 133, 177, 230, 298, 380, 485, 608, 762, 944, 1165, 1423, 1738, 2101, 2534, 3035, 3625, 4300, 5095, 5996, 7043, 8231, 9598, 11142, 12910, 14893, 17148, 19674, 22529, 25709, 29295, 33267

Offset: 0

Wesley Ivan Hurt, Jun 30 2019

nonn

Index entries for sequences related to partitions

Cf. A026813, A308926, A308927, A308928, A308930, A308931, A308932, A308933.

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

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

a(n) = A308926(n) - A308927(n) - A308928(n) - A308930(n) - A308931(n) - A308932(n) - A308933(n).

A308930 Sum of the fourth largest parts in the partitions of n into 7 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 9, 15, 22, 33, 46, 67, 91, 128, 169, 228, 297, 390, 498, 641, 806, 1018, 1263, 1569, 1921, 2358, 2856, 3460, 4151, 4978, 5915, 7030, 8287, 9763, 11425, 13357, 15526, 18030, 20825, 24027, 27597, 31660, 36167, 41276, 46921

Offset: 0

Wesley Ivan Hurt, Jun 30 2019

nonn

Index entries for sequences related to partitions

Cf. A026813, A308926, A308927, A308928, A308929, A308931, A308932, A308933.

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

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

a(n) = A308926(n) - A308927(n) - A308928(n) - A308929(n) - A308931(n) - A308932(n) - A308933(n).

A308931 Sum of the third largest parts in the partitions of n into 7 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 4, 7, 11, 19, 28, 42, 61, 88, 121, 171, 228, 307, 405, 531, 682, 880, 1111, 1403, 1750, 2174, 2670, 3281, 3983, 4827, 5807, 6965, 8291, 9861, 11641, 13718, 16080, 18804, 21881, 25423, 29392, 33921, 39001, 44753, 51163, 58411

Offset: 0

Wesley Ivan Hurt, Jun 30 2019

nonn

Index entries for sequences related to partitions

Cf. A026813, A308926, A308927, A308928, A308929, A308930, A308932, A308933.

Table[Total[IntegerPartitions[n,{7}][[All,3]]],{n,0,50}] (* Harvey P. Dale, Aug 15 2020 *)

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

a(n) = A308926(n) - A308927(n) - A308928(n) - A308929(n) - A308930(n) - A308932(n) - A308933(n).

A308932 Sum of the second largest parts in the partitions of n into 7 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 10, 15, 27, 39, 61, 87, 128, 175, 249, 331, 450, 590, 779, 998, 1292, 1628, 2063, 2567, 3198, 3923, 4827, 5854, 7106, 8538, 10253, 12197, 14516, 17126, 20197, 23658, 27683, 32197, 37422, 43247, 49931, 57382, 65865, 75275

Offset: 0

Wesley Ivan Hurt, Jun 30 2019

nonn

Index entries for sequences related to partitions

Cf. A026813, A308926, A308927, A308928, A308929, A308930, A308931, A308933.

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

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

a(n) = A308926(n) - A308927(n) - A308928(n) - A308929(n) - A308930(n) - A308931(n) - A308933(n).

A308933 Sum of the largest parts in the partitions of n into 7 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 9, 17, 27, 46, 69, 106, 153, 222, 307, 430, 577, 778, 1023, 1343, 1726, 2221, 2805, 3540, 4408, 5475, 6722, 8244, 10004, 12116, 14557, 17447, 20758, 24656, 29090, 34254, 40115, 46878, 54512, 63276, 73112, 84322, 96875, 111089

Offset: 0

Wesley Ivan Hurt, Jun 30 2019

nonn

Index entries for sequences related to partitions

Cf. A026813, A308926, A308927, A308928, A308929, A308930, A308931, A308932.

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

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

a(n) = A308926(n) - A308927(n) - A308928(n) - A308929(n) - A308930(n) - A308931(n) - A308932(n).

cp's OEIS Frontend

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

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A211862 Number of partitions of n into parts <= 7 with the property that all parts have distinct multiplicities.

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A308926 Sum of all the parts in the partitions of n into 7 parts.

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A308927 Sum of the smallest parts in the partitions of n into 7 parts.

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A308928 Sum of the sixth largest parts in the partitions of n into 7 parts.

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A308929 Sum of the fifth largest parts in the partitions of n into 7 parts.

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A308930 Sum of the fourth largest parts in the partitions of n into 7 parts.

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A308931 Sum of the third largest parts in the partitions of n into 7 parts.

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A308932 Sum of the second largest parts in the partitions of n into 7 parts.

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