A182714 -id:A182714 - cp's OEIS frontend

A206558 Number of 8's in the last section of the set of partitions of n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 8, 8, 13, 15, 23, 26, 38, 45, 63, 74, 101, 120, 160, 191, 248, 298, 383, 457, 579, 694, 868, 1038, 1287, 1536, 1890, 2251, 2746, 3267, 3962, 4698, 5665, 6706, 8043, 9496, 11337, 13354, 15876, 18657, 22089

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024792. Also number of occurrences of 8 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of eight successive terms give the partition numbers A000041.

Column 8 of A182703 and of A194812.

Cf. A000041, A135010, A138121, A182712-A182714, A206555- A206557, A206559, A206560.

A206558 = lambda n: sum(list(p).count(8) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..8} a(n+j), n >= 0.

A206559 Number of 9's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 9, 12, 15, 22, 26, 36, 45, 59, 73, 97, 117, 152, 187, 236, 289, 365, 442, 551, 671, 825, 999, 1226, 1474, 1796, 2159, 2609, 3124, 3765, 4485, 5377, 6396, 7627, 9041, 10750, 12696, 15038, 17724, 20909

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024793. Also number of occurrences of 9 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of nine successive terms give the partition numbers A000041.

Column 9 of A182703 and of A194812.

Cf. A000041, A135010, A138121, A182712-A182714, A206555-A206558, A206560.

A206559 = lambda n: sum(list(p).count(9) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..9} a(n+j), n >= 0.

A240058 Number of partitions of n such that m(1) = m(3), where m = multiplicity.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 4, 2, 8, 5, 12, 9, 21, 17, 32, 29, 52, 49, 79, 79, 123, 126, 184, 195, 278, 299, 409, 449, 603, 668, 874, 979, 1263, 1423, 1803, 2045, 2563, 2916, 3608, 4121, 5056, 5783, 7029, 8055, 9725, 11151, 13366, 15337, 18285, 20979, 24871, 28535

Offset: 1

Clark Kimberling, Mar 31 2014

			a(6) counts these 4 partitions:  6, 42, 321, 222.

Cf. A182714, A240059, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 3]], {n, 0, z}]  (* A182714 *)
t2 = Table[Count[f[n], p_ /; Count[p, 1] <= Count[p, 3]], {n, 0, z}] (* A182714(n+3) *)
t3 = Table[Count[f[n], p_ /; Count[p, 1] == Count[p, 3]], {n, 0, z}] (* A240058 *)
t4 = Table[Count[f[n], p_ /; Count[p, 1] > Count[p, 3]], {n, 0, z}]  (* A240059 *)
t5 = Table[Count[f[n], p_ /; Count[p, 1] >= Count[p, 3]], {n, 0, z}] (* A240059(n+1) *)

a(n) = A182714(n+3) - A182714(n) = A240059(n+1) - A240059(n) for n >= 0.

A240063 Number of partitions of n such that m(2) < m(3), where m = multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 5, 7, 9, 13, 18, 25, 33, 44, 58, 76, 100, 129, 165, 212, 269, 342, 431, 540, 675, 842, 1045, 1292, 1592, 1957, 2397, 2931, 3569, 4337, 5258, 6358, 7671, 9236, 11091, 13296, 15906, 18994, 22634, 26927, 31974, 37907, 44867, 53017, 62547

Offset: 0

Clark Kimberling, Mar 31 2014

			a(8) counts these 5 partitions:  53, 431, 332, 3311, 311111.

Cf. A240064, A240065, A182714, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 2] < Count[p, 3]], {n, 0, z}]  (* A240063 *)
t2 = Table[Count[f[n], p_ /; Count[p, 2] <= Count[p, 3]], {n, 0, z}] (* A240063(n+3) *)
t3 = Table[Count[f[n], p_ /; Count[p, 2] == Count[p, 3]], {n, 0, z}] (* A240064 *)
t4 = Table[Count[f[n], p_ /; Count[p, 2] > Count[p, 3]], {n, 0, z}]  (* A240065 *)
t5 = Table[Count[f[n], p_ /; Count[p, 2] >= Count[p, 3]], {n, 0, z}] (* A240065(n+2) *)
nmax = 50; CoefficientList[Series[x^3*(1 + x)/((1 + x + x^2 + x^3 + x^4) * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2021 *)

a(n) + A240064(n) + A240065(n) = A000041(n) for n >= 0.
From Vaclav Kotesovec, Oct 06 2021: (Start)
G.f.: x^3*(1 + x)/((1 + x + x^2 + x^3 + x^4) * Product_{k>=1} (1 - x^k)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (10*n*sqrt(3)).
a(n) ~ 2*A000041(n)/5. (End)

A240064 Number of partitions of n such that m(2) = m(3), where m = multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 5, 6, 8, 11, 16, 20, 26, 33, 43, 56, 71, 89, 112, 140, 177, 219, 271, 333, 411, 505, 617, 750, 912, 1105, 1339, 1612, 1940, 2327, 2789, 3334, 3978, 4733, 5625, 6670, 7903, 9338, 11021, 12980, 15273, 17940, 21043, 24640, 28822, 33661, 39273

Offset: 0

Clark Kimberling, Mar 31 2014

			a(6) counts these 5 partitions:  6, 51, 411, 321, 111111.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A240063, A240065, A182714, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 2] < Count[p, 3]], {n, 0, z}]  (* A240063 *)
t2 = Table[Count[f[n], p_ /; Count[p, 2] <= Count[p, 3]], {n, 0, z}] (* A240063(n+3) *)
t3 = Table[Count[f[n], p_ /; Count[p, 2] == Count[p, 3]], {n, 0, z}] (* A240064 *)
t4 = Table[Count[f[n], p_ /; Count[p, 2] > Count[p, 3]], {n, 0, z}]  (* A240065 *)
t5 = Table[Count[f[n], p_ /; Count[p, 2] >= Count[p, 3]], {n, 0, z}] (* A240065(n+2) *)

seq(n) = Vec((1-x^2)*(1-x^3)/((1-x^5)*eta(x + O(x*x^n)))) \\ Andrew Howroyd, Jan 01 2025

A240063(n) + a(n) + A240065(n) = A000041(n) for n >= 0.

G.f.: P(x)*(1 - x^2)*(1 - x^3)/(1 - x^5) where P(x) is the g.f. of A000041. - Andrew Howroyd, Jan 01 2025

A240065 Number of partitions of n such that m(2) > m(3), where m = multiplicity.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 4, 6, 9, 12, 17, 23, 33, 43, 59, 76, 102, 132, 173, 221, 285, 361, 462, 580, 733, 913, 1144, 1418, 1761, 2168, 2673, 3273, 4012, 4885, 5952, 7212, 8741, 10546, 12719, 15279, 18344, 21949, 26247, 31287, 37268, 44267, 52541, 62207, 73584

Offset: 0

Clark Kimberling, Mar 31 2014

			a(6) counts these 4 partitions:  42, 222, 2211, 21111.

Cf. A240063, A240064, A182714, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 2] < Count[p, 3]], {n, 0, z}]  (* A240063 *)
t2 = Table[Count[f[n], p_ /; Count[p, 2] <= Count[p, 3]], {n, 0, z}] (* A240063(n+3) *)
t3 = Table[Count[f[n], p_ /; Count[p, 2] == Count[p, 3]], {n, 0, z}] (* A240064 *)
t4 = Table[Count[f[n], p_ /; Count[p, 2] > Count[p, 3]], {n, 0, z}]  (* A240065 *)
t5 = Table[Count[f[n], p_ /; Count[p, 2] >= Count[p, 3]], {n, 0, z}] (* A240065(n+2) *)

a(n) + A240063(n) + A240064(n) = A000041(n) for n >= 0.

A194703 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (3 + m).

Original entry on oeis.org

3, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 3, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Omar E. Pol, Feb 05 2012

Sub-triangle of A182703 and also of A194812. Note that the sum of every row is also the number of partitions of 3. For further information see A182703 and A135010.

			Triangle begins:
3,
2, 1,
0, 1, 2,
1, 0, 1, 1,
0, 1, 0, 1, 1,
0, 0, 1, 0, 1, 1,
0, 0, 0, 1, 0, 1, 1,
0, 0, 0, 0, 1, 0, 1, 1,
0, 0, 0, 0, 0, 1, 0, 1, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1, 1,
...
For k = 1 and m = 1, T(1,1) = 3 because there are three parts of size 1 in the last section of the set of partitions of 4, since 3 + m = 4, so a(1) = 3.
For k = 2 and m = 1, T(2,1) = 2 because there are two parts of size 2 in the last section of the set of partitions of 4, since 3 + m = 4, so a(2) = 2.

Always the sum of row k = p(3) = A000041(3) = 3.
The first (0-10) members of this family of triangles are A023531, A129186, A194702, this sequence, A194704-A194710.
Cf. A135010, A138121, A182712-A182714, A194812.

T(k,m) = A182703(3+m,k), with T(k,m) = 0 if k > 3+m.

T(k,m) = A194812(3+m,k).

A206556 Number of 6's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 5, 4, 8, 9, 14, 16, 26, 28, 42, 50, 69, 82, 114, 133, 179, 215, 279, 335, 434, 516, 657, 789, 987, 1182, 1473, 1754, 2164, 2583, 3154, 3755, 4567, 5414, 6542, 7753, 9307, 11000, 13158, 15501, 18456, 21712, 25731, 30196, 35677

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024790. Also number of occurrences of 6 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of six successive terms give the partition numbers A000041.

Column 6 of A182703 and of A194812.

Cf. A000041, A135010, A138121, A182712-A182714, A206556-A206560.

A206556 = lambda n: sum(list(p).count(6) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..6} a(n+j), n >= 0.

A206557 Number of 7's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 23, 28, 39, 48, 64, 79, 104, 128, 165, 204, 258, 317, 399, 487, 606, 739, 912, 1105, 1356, 1637, 1994, 2400, 2906, 3485, 4199, 5016, 6015, 7164, 8553, 10151, 12076, 14286, 16930, 19974, 23588, 27749

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024791. Also number of occurrences of 7 in all partitions of n that do not contain 1 as a part. For the definition of "last section of n" see A135010. It appears that the sums of seven successive terms give the partition numbers A000041.

Column 7 of A182703 and of A194812.

Cf. A000041, A135010, A138121, A182712-A182714, A206555, A206556, A206558-A206560.

A206557 = lambda n: sum(list(p).count(7) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..7} a(n+j), n >= 0.

A207377 Triangle read by rows in which row n lists the parts of the last section of the set of partitions of n in nondecreasing order.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 4, 1, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6, 8

Offset: 1

Omar E. Pol, Feb 23 2012

Starting from the first row; it appears that the total numbers of occurrences of k in k successive rows give the sequence A000041. For more information see A182703.

			Written as a triangle:
1;
1,2;
1,1,3;
1,1,1,2,2,4;
1,1,1,1,1,2,3,5;
1,1,1,1,1,1,1,2,2,2,2,3,3,4,6;
1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,4,5,7;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4,4,5,6,8;

Alois P. Heinz, Rows n = 1..23, flattened

Triangle similar to A135010. Mirror of A207378. Row n has length A138137(n). Row sums give A138879. Right border is A000027.

Cf. A000041, A006128, A066186, A138121, A182703, A182712-A182714, A194812.

cp's OEIS Frontend

A206558 Number of 8's in the last section of the set of partitions of n.

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A206559 Number of 9's in the last section of the set of partitions of n.

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A240058 Number of partitions of n such that m(1) = m(3), where m = multiplicity.

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A240063 Number of partitions of n such that m(2) < m(3), where m = multiplicity.

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A240064 Number of partitions of n such that m(2) = m(3), where m = multiplicity.

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A240065 Number of partitions of n such that m(2) > m(3), where m = multiplicity.

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A194703 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (3 + m).

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A206556 Number of 6's in the last section of the set of partitions of n.

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A206557 Number of 7's in the last section of the set of partitions of n.

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