A027293 -id:A027293 - cp's OEIS frontend

A152722 Triangle read by rows: T(n,0) = prime(n+2), T(n,1) = 1 - T(n,0), T(n,k) = T(n-1,k-1), T(1,0) = 1 T(n,n) = -1.

-1, 1, -1, 7, -6, -1, 11, -10, -6, -1, 13, -12, -10, -6, -1, 17, -16, -12, -10, -6, -1, 19, -18, -16, -12, -10, -6, -1, 23, -22, -18, -16, -12, -10, -6, -1, 29, -28, -22, -18, -16, -12, -10, -6, -1, 31, -30, -28, -22, -18, -16, -12, -10, -6, -1, 37, -36, -30, -28, -22, -18, -16, -12, -10, -6, -1

Offset: 0

Roger L. Bagula and Alexander R. Povolotsky, Dec 11 2008

			Triangle begins as:
  -1;
   1,  -1;
   7,  -6,  -1;
  11, -10,  -6,  -1;
  13, -12, -10,  -6,  -1;
  17, -16, -12, -10,  -6,  -1;
  19, -18, -16, -12, -10,  -6,  -1;
  23, -22, -18, -16, -12, -10,  -6, -1;
  29, -28, -22, -18, -16, -12, -10, -6, -1;

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Cf. A152568, A027293.

T[n_, n_]:= -1; T[1, 0]:= 1; T[n_, 0]:= Prime[n+2]; T[n_, 1]:= 1 - Prime[n+2]; T[n_, k_]:= T[n-1, k-1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 07 2019 *)

{T(n,k) = if(k==n, -1, if(n==1 && k==0, 1, if(k==0, prime(n+2), if(k==1, 1-prime(n+2), T(n-1,k-1) ))))}; \\ G. C. Greubel, Apr 07 2019

@CachedFunction
def T(n,k):
   if k==n: return -1
   elif n==1 and k==0: return 1
   elif k==0: return nth_prime(n+2)
   elif k==1: return 1 - nth_prime(n+2)
   else: return T(n-1,k-1)
[[T(n,k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 07 2019

T(n,n) = -1, T(1,0) = 1, T(n,0) = prime(n+2), T(n,1) = 1 - prime(n+2), T(n,k) = T(n-1,k-1). - G. C. Greubel, Apr 07 2019

Edited by G. C. Greubel, Apr 07 2019

A194711 Triangle read by rows: T(n,k) = number of partitions or zones in the last section of the set of partitions of n that contains k as a part.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 1, 1, 0, 1, 7, 2, 1, 1, 0, 1, 11, 2, 2, 1, 1, 0, 1, 15, 4, 2, 1, 1, 1, 0, 1

Offset: 1

Omar E. Pol, Feb 10 2012

It appears that row n lists A000041(n)-1 together with the row n-2 of the triangle A116598, if n >= 2.

			Triangle begins:
1,
1, 1,
2, 0, 1,
3, 1, 0, 1,
5, 1, 1, 0, 1,
7, 2, 1, 1, 0, 1,
11, 2, 2, 1, 1, 0, 1,
15, 4, 2, 1, 1, 1, 0, 1,

Column 1 is A000041. Columns >= 2 are A002865.

Cf. A027293, A116598, A135010, A138121, A182703, A194812.

A228823 Triangle read by rows: T(n,k) = total number of parts in all partitions of n that contain k as a part, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 9, 5, 2, 1, 17, 9, 5, 2, 1, 27, 17, 9, 5, 2, 1, 46, 27, 17, 9, 5, 2, 1, 69, 46, 27, 17, 9, 5, 2, 1, 108, 69, 46, 27, 17, 9, 5, 2, 1, 158, 108, 69, 46, 27, 17, 9, 5, 2, 1, 234, 158, 108, 69, 46, 27, 17, 9, 5, 2, 1, 331, 234, 158, 108, 69

Offset: 1

Omar E. Pol, Sep 25 2013

Row n lists the first n elements of A093694 in decreasing order.

			Triangle begins:
1;
2,     1;
5,     2,   1;
9,     5,   2,   1;
17,    9,   5,   2,  1;
27,   17,   9,   5,  2,  1;
46,   27,  17,   9,  5,  2,  1;
69,   46,  27,  17,  9,  5,  2,  1;
108,  69,  46,  27, 17,  9,  5,  2,  1;
158, 108,  69,  46, 27, 17,  9,  5,  2,  1;
234, 158, 108,  69, 46, 27, 17,  9,  5,  2,  1;
331, 234, 158, 108, 69, 46, 27, 17,  9,  5,  2,  1;

Cf. A000041, A006128, A027293, A093694, A182701.

T(n,k) = A000041(n-k) + A006128(n-k) = A093694(n-k).

A300185 Irregular triangle read by rows: T(n, {j,k}) is the number of partitions of n that have exactly j parts equal to k; 1 <= j <= n, 1 <= k <= n.

Original entry on oeis.org

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

Offset: 1

J. Stauduhar, Feb 27 2018

Row sums = A027293.
If superfluous zeros are removed from the right side of each row, the row lengths = 1,2,1,3,1,1,4,2,... = A010766.
Sum of each N X N block of rows = 1,2,4,7,12,19,... = A000070.
The sum of the partitions of n that are over-counted in each block of N x N rows = A000070(n) - A000041(n) = A058884(n), n >= 1.
Concatenation of first row from each N X N block = A116598.
As noted by Joerg Arndt in A116598, the first row from each N X N block in reverse converges to A002865.  Two sequences emerge from alternating second rows in reverse:  for 2n, converges to even-indexed terms in A027336, and for 2n+1, converges to odd-indexed terms in A027336.
Counting the rows in each N X N block where columns j=2 > 0 and j=3 through j=n are all zeros produces A008615(n), n > 0.

			      \ j  1 2 3 4 5
     k
n
1:   1     1
2:   1     0 1
     2     1 0
3:   1     1 0 1
     2     1 0 0
     3     1 0 0
4:   1     1 1 0 1
     2     1 1 0 0
     3     1 0 0 0
     4     1 0 0 0
5:   1     2 1 1 0 1
     2     2 1 0 0 0
     3     2 0 0 0 0
     4     1 0 0 0 0
     5     1 0 0 0 0
.
.
.

J. Stauduhar, Table of n, a(n) for n = 1..10000
Jerome Kelleher and Barry O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, arXiv:0909.2331 [cs.DS], 2009-2014.
J. Stauduhar, Original Python program.

Cf. A000041, A000070, A008615, A010766, A027293, A027336, A058884, A116598.

Array[With[{s = IntegerPartitions[#]}, Table[Count[Map[Count[#, k] &, s], j], {k, #}, {j, #}]] &, 7] // Flatten (* Michael De Vlieger, Feb 28 2018 *)

Python
```
# See Stauduhar link.
```

A152721 A prime based vector recursion: a(n)={Prime[n+1],Prime[n],Prime[n-1],-Prime[n-2],...,-1,-1}.

Original entry on oeis.org

-1, 1, -1, 5, -1, -1, 7, -5, -1, -1, 11, -7, -5, -1, -1, 13, -11, -7, -5, -1, -1, 17, -13, -11, -7, -5, -1, -1, 19, -17, -13, -11, -7, -5, -1, -1, 23, -19, -17, -13, -11, -7, -5, -1, -1, 29, -23, -19, -17, -13, -11, -7, -5, -1, -1, 31, -29, -23, -19, -17, -13, -11, -7, -5

Offset: 0

Roger L. Bagula, Dec 11 2008

sign

Row sums are:

{-1, 0, 3, 0, -3, -12, -21, -36, -51, -68, -95,...}

			{-1},
{1, -1},
{5, -1, -1},
{7, -5, -1, -1},
{11, -7, -5, -1, -1},
{13, -11, -7, -5, -1, -1},
{17, -13, -11, -7, -5, -1, -1},
{19, -17, -13, -11, -7, -5, -1, -1},
{23, -19, -17, -13, -11, -7, -5, -1, -1},
{29, -23, -19, -17, -13, -11, -7, -5, -1, -1},
{31, -29, -23, -19, -17, -13, -11, -7, -5, -1, -1}

A152568, A027293

b[0] = {-1}; b[1] = {1, -1};
b[n_] := b[n] = Join[{Prime[n + 1 ]}, {-b[n - 1][[1]]}, Table[b[n - 1][[i]], {i, 2, Length[b[n - 1]]}]];
Table[b[n], {n, 0, 10}]; Flatten[%]

a(n)={Prime[n+1],Prime[n],Prime[n-1],-Prime[n-2],...,-1,-1}.

cp's OEIS Frontend

A152722 Triangle read by rows: T(n,0) = prime(n+2), T(n,1) = 1 - T(n,0), T(n,k) = T(n-1,k-1), T(1,0) = 1 T(n,n) = -1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula

Extensions

A194711 Triangle read by rows: T(n,k) = number of partitions or zones in the last section of the set of partitions of n that contains k as a part.

Views

Author

Keywords

Comments

Examples

Crossrefs

A228823 Triangle read by rows: T(n,k) = total number of parts in all partitions of n that contain k as a part, n>=1, 1<=k<=n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A300185 Irregular triangle read by rows: T(n, {j,k}) is the number of partitions of n that have exactly j parts equal to k; 1 <= j <= n, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A152721 A prime based vector recursion: a(n)={Prime[n+1],Prime[n],Prime[n-1],-Prime[n-2],...,-1,-1}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula