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.

A129867 Row sums of triangular array T: T(j,k) = k*(j-k)! for k < j, T(j,k) = 1 for k = j; 1 <= k <= j.

Original entry on oeis.org

1, 2, 5, 14, 47, 200, 1073, 6986, 53219, 462332, 4500245, 48454958, 571411271, 7321388384, 101249656697, 1502852293010, 23827244817323, 401839065437636, 7182224591785949, 135607710526966262, 2696935204638786575
Offset: 1

Views

Author

Paul Curtz, May 24 2007

Keywords

Comments

T read by rows is in A130469.
First differences are 1, 3, 9, 33, 153, 873, 5913, ... (see A007489), second differences are 2, 6, 24, 120, 720, 5040, ... (see A000142 ).
First terms of the sequences of m-th differences are 1, 2, 4, 14, 64, ... (see A055790, A047920, A068106).
Antidiagonal sums are 1, 1, 3, 8, 29, 135, ... (see A130470) with first differences 0, 2, 5, 21, 106, ... (see A130471).
Equals the row sums of irregular triangle A182961. - Paul D. Hanna, Mar 05 2012

Examples

			First seven rows of T are
[   1 ]
[   1,   1 ]
[   2,   2,   1 ]
[   6,   4,   3,   1 ]
[  24,  12,   6,   4,   1 ]
[ 120,  48,  18,   8,   5,   1 ]
[ 720, 240,  72,  24,  10,   6,   1 ]

Links

Crossrefs

Cf. A130469, A007489, A000142, A055790, A047920, A068106, A130470, A130471.
Cf. A182961.

Programs

  • Magma
    m:=21; [ &+([ k*Factorial(j-k): k in [1..j-1] ] cat [ 1 ]): j in [1..m] ]; // Klaus Brockhaus, May 28 2007

Extensions

Edited and extended by Klaus Brockhaus, May 28 2007

A130469 Triangular array read by rows: T(j,k) = k*(j-k)! for k < j, T(j,k) = 1 for k = j; 1 <= k <= j.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 4, 3, 1, 24, 12, 6, 4, 1, 120, 48, 18, 8, 5, 1, 720, 240, 72, 24, 10, 6, 1, 5040, 1440, 360, 96, 30, 12, 7, 1, 40320, 10080, 2160, 480, 120, 36, 14, 8, 1, 362880, 80640, 15120, 2880, 600, 144, 42, 16, 9, 1, 3628800, 725760, 120960, 20160, 3600
Offset: 1

Views

Author

Klaus Brockhaus, May 28 2007

Keywords

Comments

T is also defined in A129867, which gives row sums of T.

Examples

			First seven rows of T are
[ 1 ]
[ 1, 1 ]
[ 2, 2, 1 ]
[ 6, 4, 3, 1 ]
[ 24, 12, 6, 4, 1 ]
[ 120, 48, 18, 8, 5, 1 ]
[ 720, 240, 72, 24, 10, 6, 1 ]

Crossrefs

Cf. A129867, A130470 (antidiagonal sums), A130471 (first differences of antidiagonal sums).

Programs

  • Magma
    m:=11; &cat[ [ k*Factorial(j-k): k in [1..j-1] ] cat [ 1 ]: j in [1..m] ];

A130471 First differences of antidiagonal sums of triangular array T: T(j,k) = k*(j-k)! for k < j, T(j,k) = 1 for k = j; 1 <= k <= j.

Original entry on oeis.org

0, 2, 5, 21, 106, 640, 4527, 36539, 331508, 3338358, 36946489, 445724977, 5821580670, 81839381996, 1232102291651, 19778348559015, 337223021210632, 6086161135368034, 115915940643233613, 2323409451689985053
Offset: 1

Views

Author

Klaus Brockhaus, May 28 2007

Keywords

Comments

a(n) = A130470(n+1) - A130470(n).

Examples

			a(7) = A130470(8) - A130470(7) = 5302 - 775 = 4527.

Crossrefs

Cf. A130469 (T read by rows), A129867 (row sums), A130470 (antidiagonal sums).

Programs

  • Magma
    m:=21; T:=[ [ k*Factorial(j-k): k in [1..j-1] ] cat [ 1 ]: j in [1..m] ]; S:=[ &+[ T[j-k+1][k]: k in [1..(j+1) div 2] ]: j in [1..m] ]; [ S[j+1]-S[j]: j in [1..m-1] ];
Showing 1-3 of 3 results.

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

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