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 1-4 of 4 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

A130470 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

1, 1, 3, 8, 29, 135, 775, 5302, 41841, 373349, 3711707, 40658196, 486383173, 6307963843, 88147345839, 1320249637490, 21098598196505, 358321619407137, 6444482754775171, 122360423398008784, 2445769875087993837
Offset: 1

Views

Author

Klaus Brockhaus, May 28 2007

Keywords

Examples

			Antidiagonal starting at T(7,1) is 720, 48, 6, 1, so a(7) = 775.

Crossrefs

Cf. A130469 (T read by rows), A129867 (row sums of T), A130471 (first differences).

Programs

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

A182961 Triangle, read by rows, where terms in row n equal the partial sums of row n-1 with 1's inserted at positions [0,n,2n-1,3n-3,4n-6,5n-10,...,n(n+1)/2-1] for n>0, with T(0,0)=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 5, 1, 1, 2, 4, 1, 5, 8, 1, 9, 1, 14, 1, 1, 2, 4, 8, 1, 9, 14, 22, 1, 23, 32, 1, 33, 1, 47, 1, 1, 2, 4, 8, 16, 1, 17, 26, 40, 62, 1, 63, 86, 118, 1, 119, 152, 1, 153, 1, 200, 1
Offset: 0

Views

Author

Paul D. Hanna, Dec 31 2010

Keywords

Examples

			This triangle T(n,k), where k=0..n(n+1)/2 in row n>=0, begins:
1;
(1),1;
(1),1,(1),2;
(1),1,2,(1),3,(1),5;
(1),1,2,4,(1),5,8,(1),9,(1),14;
(1),1,2,4,8,(1),9,14,22,(1),23,32,(1),33,(1),47;
(1),1,2,4,8,16,(1),17,26,40,62,(1),63,86,118,(1),119,152,(1),153,(1),200;
(1),1,2,4,8,16,32,(1),33,50,76,116,178,(1),179,242,328,446,(1),447,566,718,(1),719,872,(1),873,(1),1073;
...
where row n is equal to the partial sums of terms in row n-1, with 1's inserted at positions [0,n,2n-1,3n-3,4n-6,5n-10,...,n(n+1)/2-1].
The row sums and rightmost border form sequence A129867, which equals the row sums of triangle A130469.
Triangle A130469 begins:
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; ...
which has the same row sums as this triangle.

Links

Crossrefs

Cf. A129867, A130469; variant: A131338.

Programs

  • PARI
    {T(n,k)=local(A=[1],B); for(m=0,n, t=0;B=[];
for(j=0,#A-1, if(j==t*m-t*(t+1)/2, t+=1;B=concat(B,1)); B=concat(B,A[j+1]));
A=Vec( Ser(B)/(1-x+O(x^#B)) ) ); if(k+1>#A, 0, B[k+1])}
for(n=0,12,for(k=0,n*(n+1)/2,print1(T(n, k), ", ")); print(""))

Formula

Row sums equal A129867;
n-th row sum = 1 + Sum_{k=1..n} k*(n-k+1)!.
T(n,n(n+1)/2) = A129867(n) for n>0, with T(0,0) = 1.
Showing 1-4 of 4 results.

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