A194812 -id:A194812 - cp's OEIS frontend

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

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

A194705 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (5 + m).

Original entry on oeis.org

7, 4, 3, 2, 2, 3, 1, 1, 3, 2, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 2, 0, 1, 0, 1, 1, 2, 2, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 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 5. For further information see A182703 and A135010.

			Triangle begins:
  7,
  4, 3,
  2, 2, 3,
  1, 1, 3, 2,
  0, 1, 1, 2, 3,
  1, 0, 1, 1, 2, 2,
  0, 1, 0, 1, 1, 2, 2,
  ...
For k = 1 and m = 1: T(1,1) = 7 because there are seven parts of size 1 in the last section of the set of partitions of 6, since 5 + m = 6, so a(1) = 7.
For k = 2 and m = 1: T(2,1) = 4 because there are four parts of size 2 in the last section of the set of partitions of 6, since 5 + m = 6, so a(2) = 4.

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

Always the sum of row k = p(5) = A000041(5) = 7.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194704, this sequence, A194706-A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=5); 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(5+m,k), with T(k,m) = 0 if k > 5+m.

T(k,m) = A194812(5+m,k).

Terms a(29) and beyond from Andrew Howroyd, Feb 19 2020

A194706 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (6 + m).

Original entry on oeis.org

11, 3, 8, 2, 3, 6, 1, 3, 2, 5, 1, 1, 2, 3, 4, 0, 1, 1, 2, 2, 5, 1, 0, 1, 1, 2, 2, 4, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4

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 6. For further information see A182703 and A135010.

			Triangle begins:
  11,
   3, 8,
   2, 3, 6,
   1, 3, 2, 5,
   1, 1, 2, 3, 4,
   0, 1, 1, 2, 2, 5,
  ...
For k = 1 and m = 1: T(1,1) = 11 because there are 11 parts of size 1 in the last section of the set of partitions of 7, since 6 + m = 7, so a(1) = 11.
For k = 2 and m = 1: T(2,1) = 3 because there are three parts of size 2 in the last section of the set of partitions of 7, since 6 + m = 7, so a(2) = 3.

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

Always the sum of row k = p(6) = A000041(6) = 11.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194705, this sequence, A194707-A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=6); 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(6+m,k), with T(k,m) = 0 if k > 6+m.

T(k,m) = A194812(6+m,k).

Terms a(22) and beyond from Andrew Howroyd, Feb 19 2020

A194707 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (7 + m).

Original entry on oeis.org

15, 8, 7, 3, 6, 6, 3, 2, 5, 5, 1, 2, 3, 4, 5, 1, 1, 2, 2, 5, 4, 0, 1, 1, 2, 2, 4, 5, 1, 0, 1, 1, 2, 2, 4, 4, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4

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 7. For further information see A182703 and A135010.

			Triangle begins:
  15,
   8, 7,
   3, 6, 6,
   3, 2, 5, 5,
  ...
For k = 1 and m = 1: T(1,1) = 15 because there are 15 parts of size 1 in the last section of the set of partitions of 8, since 7 + m = 8, so a(1) = 15.
For k = 2 and m = 1: T(2,1) = 8 because there are eight parts of size 2 in the last section of the set of partitions of 8, since 7 + m = 8, so a(2) = 8.

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

Always the sum of row k = p(7) = A000041(7) = 15.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194706, this sequence, A194708-A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=7); 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(7+m,k), with T(k,m) = 0 if k > 7+m.

T(k,m) = A194812(7+m,k).

Terms a(11) and beyond from Andrew Howroyd, Feb 19 2020

A194708 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (8 + m).

Original entry on oeis.org

22, 7, 15, 6, 6, 10, 2, 5, 5, 10, 2, 3, 4, 5, 8, 1, 2, 2, 5, 4, 8, 1, 1, 2, 2, 4, 5, 7, 0, 1, 1, 2, 2, 4, 4, 8, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7

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 8. For further information see A182703 and A135010.

			Triangle begins:
  22,
   7, 15,
   6,  6, 10,
   2,  5,  5, 10,
   2,  3,  4,  5,  8,
   ...
For k = 1 and m = 1: T(1,1) = 22 because there are 22 parts of size 1 in the last section of the set of partitions of 9, since 8 + m = 9, so a(1) = 22.
For k = 2 and m = 1: T(2,1) = 7 because there are seven parts of size 2 in the last section of the set of partitions of 9, since 8 + m = 9, so a(2) = 7.

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

Always the sum of row k = p(8) = A000041(8) = 22.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194707, this sequence, A194709, A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=8); 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(8+m,k), with T(k,m) = 0 if k > 8+m.

T(k,m) = A194812(8+m,k).

Terms a(11) and beyond from Andrew Howroyd, Feb 19 2020

A194709 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (9 + m).

Original entry on oeis.org

30, 15, 15, 6, 10, 14, 5, 5, 10, 10, 3, 4, 5, 8, 10, 2, 2, 5, 4, 8, 9, 1, 2, 2, 4, 5, 7, 9, 1, 1, 2, 2, 4, 4, 8, 8, 0, 1, 1, 2, 2, 4, 4, 7, 9, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 4, 7, 8

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 9. For further information see A182703 and A135010.

			Triangle begins:
  30;
  15, 15;
   6, 10, 14;
   5,  5, 10, 10;
   3,  4,  5,  8, 10;
   2,  2,  5,  4,  8, 9;
  ...
For k = 1 and  m = 1; T(1,1) = 30 because there are 30 parts of size 1 in the last section of the set of partitions of 10, since 9 + m = 10, so a(1) = 30. For k = 2 and m = 1; T(2,1) = 15 because there are 15 parts of size 2 in the last section of the set of partitions of 10, since 9 + m = 10, so a(2) = 15.

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

Always the sum of row k = p(9) = A000041(n) = 30.
The first (0-10) members of this family of triangles are A023531, A129186, A194702-A194708, this sequence, A194710.
Cf. A135010, A138121, A194812.

P(n)={my(M=matrix(n,n), d=9); 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(9+m,k), with T(k,m) = 0 if k > 9+m.

T(k,m) = A194812(9+m,k).

Terms a(7) 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.

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).

cp's OEIS Frontend

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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A194705 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (5 + m).

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A194706 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (6 + m).

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A194707 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (7 + m).

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A194708 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (8 + m).

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A194709 Triangle read by rows: T(k,m) = number of occurrences of k in the last section of the set of partitions of (9 + m).

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