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 1-3 of 3 results.

A135902 Triangle T, read by rows, where column k of T = column 0 of T^(k+1) for k>0, with column 0 of T = column 0 of T^3 shift right.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 15, 8, 3, 1, 102, 47, 15, 4, 1, 860, 356, 102, 24, 5, 1, 8548, 3252, 860, 186, 35, 6, 1, 97094, 34448, 8548, 1736, 305, 48, 7, 1, 1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1, 17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1
Offset: 0

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Author

Paul D. Hanna, Dec 18 2007

Keywords

Comments

This is a variant of permutation triangle P = A094587, A094587(n,k) = n!/k!, which may be defined by: triangular matrix P where column k of P = column 0 of P^(k+1), with column 0 of P = column 0 of P^2 shift right.

Examples

			Triangle T begins:
1;
1, 1;
3, 2, 1;
15, 8, 3, 1;
102, 47, 15, 4, 1;
860, 356, 102, 24, 5, 1;
8548, 3252, 860, 186, 35, 6, 1;
97094, 34448, 8548, 1736, 305, 48, 7, 1;
1234324, 412546, 97094, 18754, 3130, 465, 63, 8, 1;
17302880, 5488222, 1234324, 228658, 36630, 5212, 672, 80, 9, 1; ...
where column k of T = column 0 of T^(k+1)
with column 0 of T = column 0 of T^3 shift right.
Matrix square of T, T^2, begins:
1;
2, 1;
8, 4, 1;
47, 22, 6, 1;
356, 156, 42, 8, 1;
3252, 1343, 351, 68, 10, 1;
34448, 13493, 3415, 656, 100, 12, 1; ...
where column 0 of T^2 = column 1 of T.
Matrix cube of T, T^3, begins:
1;
3, 1;
15, 6, 1;
102, 42, 9, 1;
860, 351, 81, 12, 1;
8548, 3415, 807, 132, 15, 1;
97094, 37795, 8967, 1530, 195, 18, 1; ...
where column 0 of T^3 = column 2 of T = column 0 of T shift left;
also, column 1 of T^3 = column 2 of T^2.
		

Crossrefs

Cf. columns: A135903, A135904, A135905; variants: A091351, A094587.

Programs

  • PARI
    T(n, k)=if(k>n || n<0 || k<0, 0, if(k==n,1,if(k==0, T(n+1,2), sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1))); ); )

Formula

Column k of T^(j+1) = column j of T^(k+1) for j>=0, k>=0.

A135904 Column 1 of triangle A135902.

Original entry on oeis.org

1, 2, 8, 47, 356, 3252, 34448, 412546, 5488222, 80066096, 1268400628, 21651578958, 395749634342, 7705289481276, 159106051976460, 3471184678351602, 79751883928130922, 1924103020594546192, 48622055377338993264
Offset: 0

Views

Author

Paul D. Hanna, Dec 18 2007

Keywords

Crossrefs

A135905 Column 3 of triangle A135902.

Original entry on oeis.org

1, 4, 24, 186, 1736, 18754, 228658, 3092316, 45802294, 735881882, 12727541778, 235515880084, 4638914346322, 96840935881898, 2134717733185118, 49529467079270060, 1206148832249659798, 30751284267472226682
Offset: 0

Views

Author

Paul D. Hanna, Dec 18 2007

Keywords

Crossrefs

Showing 1-3 of 3 results.