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.

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A128568 Column 1 of triangle A128567.

Original entry on oeis.org

1, 6, 31, 133, 587, 2531, 10950, 47185, 203704, 879711, 3804530, 16464710, 71312805, 309083291, 1340546867, 5817555402, 25258769216, 109711224970, 476675868834, 2071569641859, 9004430215111, 39144480326143, 170184867215647, 739924236443359, 3217001700174226
Offset: 0

Views

Author

Paul D. Hanna, Mar 12 2007

Keywords

Comments

A128567 is the matrix square of Parker's partition triangle A047812.

Crossrefs

Cf. A007042, A047812, A128567, A128569 (column 2), A128602 (row sums).

Programs

  • PARI
    {a(n)=local(M);M=matrix(n+2,n+2,r,c,if(r
    				

Formula

a(n) = Sum_{s=1..n+1} A047812(n+2,s)*A047812(s+1,1) = Sum_{s=1..n+1} A047812(n+2,s)*A007042(s+1) for n >= 0. - Petros Hadjicostas, May 31 2020

A128569 Column 2 of triangle A128567.

Original entry on oeis.org

1, 14, 117, 813, 4871, 27743, 151208, 804065, 4185683, 21472005, 108766010, 545507633, 2712801330, 13394412999, 65722444172, 320721839860, 1557502222385, 7530671086667, 36267851679585, 174038009185816, 832392015517829, 3969017685816667, 18871416851149078
Offset: 0

Views

Author

Paul D. Hanna, Mar 12 2007

Keywords

Comments

A128567 is the matrix square of Parker's partition triangle A047812.

Crossrefs

Cf. A047812, A128567, A128568 (column 1), A128602 (row sums).

Programs

  • PARI
    {a(n)=local(M);M=matrix(n+3,n+3,r,c,if(r
    				

Formula

a(n) = Sum_{s=2..n+2} A047812(n+3,s)*A047812(s+1,2) for n >= 0. - Petros Hadjicostas, May 31 2020

A128602 Row sums of triangle A128567.

Original entry on oeis.org

1, 3, 12, 60, 315, 1869, 11472, 74797, 502908, 3505031, 24973089, 182208083, 1352620790, 10207771213, 78082422354, 604699597868, 4733082767467, 37406134058641, 298165102770381, 2395219866441531, 19376637845028027, 157757577529194488, 1291926016200778464
Offset: 1

Views

Author

Paul D. Hanna, Mar 12 2007

Keywords

Comments

A128567 is the matrix square of Parker's partition triangle A047812.

Crossrefs

Cf. A000108, A047812, A128567, A128568 (column 1), A128569 (column 2).

Programs

  • PARI
    {a(n)=local(M);M=matrix(n+1,n+1,r,c,if(rPetros Hadjicostas, May 31 2020

Formula

a(n) = Sum_{s=0..n-1} A047812(n,s)*A000108(s+1) for n >= 1. - Petros Hadjicostas, May 31 2020

Extensions

Offset changed by Petros Hadjicostas, May 31 2020 to agree with A128567

A128562 Triangle, read by rows, where T(n,k) is the coefficient of q^((n+1)*k) in the q-binomial coefficient [2*n+1, n] for n >= k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 12, 6, 1, 1, 10, 29, 29, 10, 1, 1, 14, 61, 94, 61, 14, 1, 1, 21, 120, 263, 263, 120, 21, 1, 1, 29, 222, 645, 910, 645, 222, 29, 1, 1, 41, 392, 1468, 2724, 2724, 1468, 392, 41, 1, 1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1
Offset: 0

Views

Author

Paul D. Hanna, Mar 10 2007

Keywords

Comments

Row sums equal a shifted version of A003239 (number of rooted planar trees with n non-root nodes). Column 1 is a shifted version of A000065 (-1 + number of partitions of n). Column 2 is a shifted version of A128563. This array is a variant of triangles A128545 and A047812 (Parker's partition triangle).

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   4,    1;
  1,  6,  12,    6,    1;
  1, 10,  29,   29,   10,    1;
  1, 14,  61,   94,   61,   14,    1;
  1, 21, 120,  263,  263,  120,   21,    1;
  1, 29, 222,  645,  910,  645,  222,   29,   1;
  1, 41, 392, 1468, 2724, 2724, 1468,  392,  41,  1;
  1, 55, 669, 3113, 7352, 9686, 7352, 3113, 669, 55, 1;
  ...
		

Crossrefs

Cf. A000065 (column 1), A003239 (row sums), A128563 (column 2).
Variants are A047812 and A128545.

Programs

  • PARI
    T(n,k)=if(n
    				

Formula

T(n,k) = [q^((n+1)*k)] Product_{j=n+1..2*n+1}(1-q^j) / Product_{j=1..n+1}(1-q^j).

Extensions

Minor edits by Petros Hadjicostas, Jun 01 2020
Previous Showing 11-14 of 14 results.