cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A096807 Row sums of triangle A096806, in which the binomial transform of the n-th row lists the m-dimensional partitions of n, for n>=1 and m>=0.

Original entry on oeis.org

1, 2, 4, 10, 26, 78, 246, 844, 3062, 11782, 47664, 202254, 896462, 4139514
Offset: 1

Views

Author

Paul D. Hanna, Jul 19 2004

Keywords

Crossrefs

Cf. A096751, A096806.

A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 140, 86, 22, 1, 1, 1, 9, 36, 105, 216, 326, 307, 160, 30, 1, 1, 1, 10, 45, 148, 357, 657, 835, 684, 282, 42, 1
Offset: 0

Views

Author

Paul D. Hanna, Jul 07 2004

Keywords

Comments

Main diagonal forms A096752. Antidiagonal sums form A096753. Row with index n lists the row sums of the n-th matrix power of triangle A096651, for n>=0.

Examples

			n-th row lists n-dimensional partitions; table begins with n=0:
  [1,1,1,1,1,1,1,1,1,1,1,1,...],
  [1,1,2,3,5,7,11,15,22,30,42,56,...],
  [1,1,3,6,13,24,48,86,160,282,500,859,...],
  [1,1,4,10,26,59,140,307,684,1464,3122,...],
  [1,1,5,15,45,120,326,835,2145,5345,...],
  [1,1,6,21,71,216,657,1907,5507,15522,...],
  [1,1,7,28,105,357,1197,3857,12300,38430,...],
  [1,1,8,36,148,554,2024,7134,24796,84625,...],
  [1,1,9,45,201,819,3231,12321,46209,170370,...],
  [1,1,10,55,265,1165,4927,20155,80920,...],...
Array begins:
      k=0:  k=1:  k=2:  k=3:  k=4:  k=5:  k=6:  k=7:  k=8:
  n=0:  1     1     1     1     1     1     1     1     1
  n=1:  1     1     2     3     5     7    11    15    22
  n=2:  1     1     3     6    13    24    48    86   160
  n=3:  1     1     4    10    26    59   140   307   684
  n=4:  1     1     5    15    45   120   326   835  2145
  n=5:  1     1     6    21    71   216   657  1907  5507
  n=6:  1     1     7    28   105   357  1197  3857 12300
  n=7:  1     1     8    36   148   554  2024  7134 24796
  n=8:  1     1     9    45   201   819  3231 12321 46209
  n=9:  1     1    10    55   265  1165  4927 20155 80920

References

  • G. E. Andrews, The Theory of Partitions, Add.-Wes. 1976, pp. 189-197.

Links

Crossrefs

Rows: A000012 (n=0), A000041 (n=1), A000219 (n=2), A000293 (n=3), A000334 (n=4), A000390 (n=5), A000416 (n=6), A000427 (n=7), A179855 (n=8).
Columns: A008778 (k=4), A008779 (k=5), A042984 (k=6).
Cf. A096651, A096752, A096753.
Cf. A096806.
Cf. A042984.
Cf. A144150, A213427, A290353, A323718, A323719.

Programs

  • Mathematica
    trans[x_]:=If[x=={},{},Transpose[x]];
levptns[n_,k_]:=If[k==1,IntegerPartitions[n],Join@@Table[Select[Tuples[levptns[#,k-1]&/@y],And@@(GreaterEqual@@@trans[Flatten/@(PadRight[#,ConstantArray[n,k-1]]&/@#)])&],{y,IntegerPartitions[n]}]];
Table[If[sum==k,1,Length[levptns[k,sum-k]]],{sum,0,10},{k,0,sum}] (* Gus Wiseman, Jan 27 2019 *)

Formula

T(0, n)=T(n, 0)=T(n, 1)=1 for n>=0.
Inverse binomial transforms of the columns is given by triangle A096806.

A119271 Triangle: number of exactly (m-1)-dimensional partitions of n, for n >= 1, m >= 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 9, 18, 10, 1, 0, 1, 13, 44, 49, 15, 1, 0, 1, 20, 97, 172, 110, 21, 1, 0, 1, 28, 195, 512, 550, 216, 28, 1, 0, 1, 40, 377, 1370, 2195, 1486, 385, 36, 1, 0, 1, 54, 694, 3396, 7603, 7886, 3514, 638, 45, 1, 0, 1, 75, 1251, 7968
Offset: 1

Views

Author

Franklin T. Adams-Watters, May 11 2006

Keywords

Comments

The partition of 1 is considered to be dimension -1 by convention.

Examples

			Table starts:
  1,
  0,1,
  0,1,1,
  0,1,3,1,
  0,1,5,6,1,
  ...

Links

Crossrefs

Cf. A119270, A096806. Column 1 is A007042.

Formula

a(n,m) = A096806(n,m-1)-a(n,m-1). Binomial transform of n-th row lists the (m-1) dimensional partitions of n.

A116673 Row sums of triangle A116672, in which the binomial transform of the n-th row lists the Euler transform of the n-th sequence in A007318 (Pascal's Triangle).

Original entry on oeis.org

1, 2, 4, 10, 26, 80, 262, 950
Offset: 1

Views

Author

Alford Arnold, Feb 22 2006

Keywords

Comments

A116673 is to A096807 as Table A116672 is to Table A096806. The difference between the two tables is of historical interest. (cf. A096751 and A007326).

Examples

			A116672 begins
1; 1,1; 1,2,1; 1,4,4,1; 1,6,11,7,1; 1,10,27,29,12,1; 1,14,57,96,72,21,1; 1,21,117,277,319,176,38,1; . . . so
A116673 begins 1 2 4 10 26 80 262 950 ...

Crossrefs

Cf. A007318, A007326, A096751, A096806, A096807, A116672.

A167366 Triangle read by rows, 2*A047999 - A097806 (signed) = twice Sierpinski's gasket - the signed pair sum operator.

Original entry on oeis.org

1, 3, 1, 2, 1, 1, 2, 2, 3, 1, 2, 0, 0, 1, 1, 2, 2, 0, 0, 3, 1, 2, 0, 2, 0, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 3, 1, 2, 0, 2, 0, 0, 0, 0, 0, 2, 1, 1, 2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 3, 1, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1
Offset: 0

Views

Author

Gary W. Adamson & Mats Granvik, Nov 01 2009

Keywords

Comments

Row sums = A167275: (1, 4, 4, 8, 4, 8, 8, 16,...).

Examples

			First few rows of the triangle =
1;
3, 1;
2, 1, 1;
2, 2, 3, 1;
2, 0, 0, 1, 1;
2, 2, 0, 0, 3, 1;
2, 0, 2, 0, 2, 1, 1;
2, 2, 2, 2, 2, 2, 3, 1;
2, 0, 0, 0, 0, 0, 0, 1, 1;
2, 2, 0, 0, 0, 0, 0, 0, 3, 1;
2, 0, 2, 0, 0, 0, 0, 0, 2, 1, 1;
2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 3, 1;
2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1;
2, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 1;
2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 1, 1;
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1;
2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1;
2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 1;
2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 1;
...

Crossrefs

Cf. A047999, A097806

Formula

Triangle read by rows, 2*A047999 - A096806, the pair sum operator.
A096806 is signed, rightmost diagonal = (+,+,+,...); adjacent diagonal is
signed (-,-,-,...).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.