A182712 -id:A182712 - cp's OEIS frontend

A206555 Number of 5's in the last section of the set of partitions of n.

0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 4, 5, 8, 10, 15, 18, 26, 32, 44, 56, 73, 92, 120, 149, 193, 238, 302, 373, 469, 576, 716, 876, 1081, 1316, 1615, 1954, 2383, 2875, 3483, 4188, 5048, 6043, 7253, 8653, 10341, 12293, 14634, 17340, 20567, 24300, 28717, 33830

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024789. Also number of occurrences of 5 in all partitions of n that do not contain 1 as a part. It appears that the sum of five successive terms gives the partition numbers A000041 (see A182703 and A194812).

Column 5 of A182703 and of A194812.

Cf. A135010, A138121, A182712-A182714.

A206555 = lambda n: sum(list(p).count(5) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..5} a(n+j), n >= 0.

More terms from Alois P. Heinz, Feb 20 2012

A206560 Number of 10'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, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 14, 22, 25, 36, 43, 59, 70, 95, 113, 150, 179, 232, 278, 356, 426, 537, 644, 803, 960, 1189, 1417, 1739, 2072, 2523, 2999, 3631, 4304, 5181, 6130, 7342, 8662, 10330, 12159, 14437, 16958

Offset: 1

Omar E. Pol, Feb 09 2012

nonn

Zero together with the first differences of A024794. Also number of occurrences of 10 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 ten successive terms give the partition numbers A000041.

Column 10 of A182703 and of A194812.

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

A206560 = lambda n: sum(list(p).count(10) for p in Partitions(n) if 1 not in p)

It appears that A000041(n) = Sum_{j=1..10} a(n+j), n >= 0.

A194704 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (4 + m).

Original entry on oeis.org

5, 1, 4, 1, 2, 2, 0, 1, 1, 3, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2

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 4. For further information see A182703 and A135010.

			Triangle begins:
  5,
  1, 4,
  1, 2, 2,
  0, 1, 1, 3,
  1, 0, 1, 1, 2,
  ...
For k = 1 and m = 1: T(1,1) = 5 because there are five parts of size 1 in the last section of the set of partitions of 5, since 4 + m = 5, so a(1) = 5.
For k = 2 and m = 1: T(2,1) = 1 because there is only one part of size 2 in the last section of the set of partitions of 5, since 4 + m = 5, so a(2) = 1.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Always the sum of row k = p(4) = A000041(4) = 5.
The first (0-10) members of this family of triangles are A023531, A129186, A194702, A194703, this sequence, A194705-A194710.
Cf. A135010, A138121, A182712-A182714, A194812.

P(n)={my(M=matrix(n,n), d=4); M[1,1]=numbpart(d); for(m=1, n, forpart(p=m+d, for(k=1, #p, my(t=p[k]); if(t<=n && m<=t, M[t, m]++)), [2, m+d])); M}
{ my(T=P(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

T(k,m) = A182703(4+m,k), with T(k,m) = 0 if k > 4+m.

T(k,m) = A194812(4+m,k).

Terms a(16) and beyond from Andrew Howroyd, Feb 19 2020

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

Original entry on oeis.org

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.

A182717 Number of 2's in all partitions of 2n+1 that do not contain 1 as a part.

Original entry on oeis.org

0, 0, 1, 3, 7, 15, 29, 53, 94, 160, 265, 430, 683, 1066, 1640, 2487, 3725, 5519, 8092, 11752, 16922, 24167, 34254, 48213, 67409, 93661, 129378, 177720, 242841, 330172, 446772, 601810, 807153, 1078081, 1434250, 1900860, 2510097, 3303003, 4331767, 5662539

Offset: 0

Omar E. Pol, Dec 03 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

A182743. Bisection of A182712.

b:= proc(n,i) option remember; local r;
      if n<=0 or i<2 then 0
    elif i=2 then `if`(irem(n,2,'r')=0,r,0)
    else b(n,i-1) +b(n-i,i)
      fi
    end:
a:= n-> b(2*n+1, 2*n+1):
seq(a(n), n=0..45);  # Alois P. Heinz, Dec 03 2010

b[n_, i_] := b[n, i] = If[n <= 0 || i < 2, 0, If[i == 2, If[Mod[n, 2] == 0, Quotient[n, 2], 0], b[n, i-1] + b[n-i, i]]];
a[n_] := b[2n+1, 2n+1];
a /@ Range[0, 45] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)
Table[Count[Flatten[Select[IntegerPartitions[2 n+1],FreeQ[#,1]&]],2],{n,0,40}] (* Harvey P. Dale, Jan 28 2022 *)

More terms from Alois P. Heinz, Dec 03 2010

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

A206555 Number of 5's in the last section of the set of partitions of n.

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A206560 Number of 10's in the last section of the set of partitions of n.

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A194704 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (4 + m).

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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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A182717 Number of 2's in all partitions of 2n+1 that do not contain 1 as a part.

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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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