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.

A115084 Row sums of triangle A115080.

Original entry on oeis.org

1, 2, 6, 20, 82, 397, 2186, 13241, 86578, 603249, 4440800, 34313272, 276904847, 2324369733, 20227700385, 181987602305, 1688734327673, 16129293322211, 158281104406067, 1593383284029479, 16431757177327175, 173372313294822983
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Crossrefs

Cf. A115080, A115081 (column 0), A115082 (column 1), A115083 (column 2); A115089.

A115085 Triangle, read by rows, where T(n,k) equals the dot product of the vector of terms in row n-1 from T(n-1,k) to T(n-1,n-1) with the vector of terms in column k+1 from T(k+1,k+1) to T(n,k+1): T(n,k) = Sum_{j=0..n-k-1} T(n-1,j+k)*T(j+k+1,k+1) for n>k+1>0, with T(n,n) = 1 and T(n,n-1) = n (n>=1).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 12, 5, 3, 1, 58, 21, 7, 4, 1, 321, 102, 32, 9, 5, 1, 1963, 579, 158, 45, 11, 6, 1, 13053, 3601, 933, 226, 60, 13, 7, 1, 92946, 24426, 5939, 1395, 306, 77, 15, 8, 1, 702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1, 5599204, 1359906, 306070
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Comments

Triangle A115080 is the dual of this triangle.

Examples

			T(n,k)=[T(n-1,k),T(n-1,k+1),..,T(n-1,n-1)]*[T(k+1,k+1),T(k+2,k+1),..,T(n,k+1)]:
12 = [3,2,1]*[1,2,5] = 3*1 + 2*2 + 1*5;
21 = [5,3,1]*[1,3,7] = 5*1 + 3*3 + 1*7;
102 = [21,7,4,1]*[1,3,7,32] = 21*1 + 7*3 + 4*7 + 1*32;
158 = [32,9,5,1]*[1,4,9,45] = 32*1 + 9*4 + 5*9 + 1*45.
Triangle begins:
1;
1, 1;
3, 2, 1;
12, 5, 3, 1;
58, 21, 7, 4, 1;
321, 102, 32, 9, 5, 1;
1963, 579, 158, 45, 11, 6, 1;
13053, 3601, 933, 226, 60, 13, 7, 1;
92946, 24426, 5939, 1395, 306, 77, 15, 8, 1;
702864, 176858, 41385, 9097, 1977, 398, 96, 17, 9, 1;
5599204, 1359906, 306070, 65310, 13195, 2691, 502, 117, 19, 10, 1;
46746501, 10996740, 2403792, 494022, 97701, 18353, 3549, 618, 140, 21, 11, 1;
407019340, 93136545, 19799468, 3970878, 755834, 140178, 24691, 4563, 746, 165, 23, 12, 1; ...
		

Crossrefs

Cf. A115086 (column 0), A115087 (column 1), A115088 (column 2), A115089 (row sums); A115080 (dual triangle).

Programs

  • PARI
    {T(n,k)=if(n==k,1,if(n==k+1,n, sum(j=0,n-k-1,T(n-1,j+k)*T(j+k+1,k+1))))}
    for(n=0,12,for(k=0,n, print1(T(n,k),", "));print(""))

A115086 Column 0 of triangle A115085.

Original entry on oeis.org

1, 1, 3, 12, 58, 321, 1963, 13053, 92946, 702864, 5599204, 46746501, 407019340, 3682922657, 34519445611, 334288656446, 3336972646500, 34270606413428, 361466138304372, 3909661505975148, 43305438098350844, 490632571238216052
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Crossrefs

Cf. A115085, A115087 (column 1), A115088 (column 2), A115089 (row sums); A115081.

A115087 Column 1 of triangle A115085.

Original entry on oeis.org

1, 2, 5, 21, 102, 579, 3601, 24426, 176858, 1359906, 10996740, 93136545, 821734072, 7529402477, 71404315785, 699210393923, 7053151126260, 73158494370744, 778922147312076, 8500606925203992, 94961268583354364, 1084632255967292304
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Crossrefs

Cf. A115085, A115086 (column 0), A115088 (column 2), A115089 (row sums); A115082.

A115088 Column 2 of triangle A115085.

Original entry on oeis.org

1, 3, 7, 32, 158, 933, 5939, 41385, 306070, 2403792, 19799468, 170676213, 1530176012, 14235081941, 136908376331, 1358591876186, 13876456179864, 145643421890800, 1568070473632644, 17295212337571836, 195161233968831068
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2006

Keywords

Crossrefs

Cf. A115085, A115086 (column 0), A115087 (column 1), A115089 (row sums); A115083.
Showing 1-5 of 5 results.