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 1-6 of 6 results.

A136741 Diagonal of square array A136737, one place above the main diagonal.

Original entry on oeis.org

1, 3, 22, 250, 3810, 72492, 1649634, 43626510, 1313526375, 44332221175, 1657043432088, 67929461003560, 3029864359322346, 146058681728370600, 7566706624571096610, 419220650458638848514, 24733868801871384287055
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2008

Keywords

Crossrefs

Cf. A136737; A121413, A121417, A121422; A121427.

A121422 Column 1 of triangle A121420.

Original entry on oeis.org

1, 3, 15, 118, 1317, 19311, 352636, 7722840, 197354133, 5764942816, 189460961985, 6917588290044, 277765971072770, 12163275845132298, 576793897136731632, 29444368084753254610, 1609805318425385690712, 93843785859803533422675
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2006

Keywords

Comments

Also column 2 of square array A136737.
A121420 is the matrix cube of triangle A121412; row n of triangle T=A121412 equals row (n-1) of T^(n+1) with an appended '1'.

Crossrefs

Cf. A121420 (triangle); other columns: A121421, A121423.
Cf. A136737; A121413, A121417; A121427, A136741.

Programs

  • PARI
    {a(n)=local(A=Mat(1), B); for(m=1, n+2, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i,j]=1, B[i, j]=(A^i)[i-1, j]); )); A=B); return((A^3)[n+2, 2])}

Extensions

Edited by N. J. A. Sloane, Oct 30 2008 at the suggestion of R. J. Mathar

A121417 Column 1 of triangle A121416.

Original entry on oeis.org

1, 2, 9, 69, 769, 11346, 208914, 4613976, 118840164, 3496297632, 115638728395, 4246267163601, 171369282105510, 7538270885559264, 358926669220446804, 18389706733665138450, 1008742283718489346668, 58981158542987625464424
Offset: 0

Views

Author

Paul D. Hanna, Aug 22 2006

Keywords

Comments

Also column 1 of square array A136737.
A121416 is the matrix square of triangle A121412; row n of triangle T=A121412 equals row (n-1) of T^(n+1) with an appended '1'.

Crossrefs

Cf. A121416 (triangle); other columns: A121418, A121419.
Cf. A136737; A121413, A121422; A121427, A136741.

Programs

  • PARI
    {a(n)=local(A=Mat(1), B); for(m=1, n+2, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i,j]=1, B[i, j]=(A^i)[i-1, j]); )); A=B); return((A^2)[n+2, 2])}

Extensions

Edited by N. J. A. Sloane, Oct 30 2008 at the suggestion of R. J. Mathar

A136730 Square array, read by antidiagonals, where T(n,k) = T(n,k-1) + T(n-1,k+n-1) for n>0, k>0, such that T(n,0) = T(n-1,n-1) for n>0 with T(0,k)=1 for k>=0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 9, 5, 3, 1, 70, 23, 9, 4, 1, 795, 175, 43, 14, 5, 1, 11961, 1935, 324, 70, 20, 6, 1, 224504, 28432, 3510, 527, 105, 27, 7, 1, 5051866, 523290, 50528, 5624, 795, 149, 35, 8, 1, 132523155, 11587072, 913377, 79553, 8396, 1140, 203, 44, 9, 1, 3969912160
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2008

Keywords

Examples

			The square array begins:
1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...];
(1), 2, 3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,...;
(2,5), 9, 14,20,27,35,44,54,65,77,90,104,119,135,152,170,189,209,230,..;
(9,23,43), 70, 105,149,203,268,345,435,539,658,793,945,1115,1304,1513,.;
(70,175,324,527), 795, 1140,1575,2114,2772,3565,4510,5625,6929,8442,...;
(795,1935,3510,5624,8396), 11961, 16471,22096,29025,37467,47652,59832,..;
(11961,28432,50528,79553,117020,164672), 224504, 298786,390087,501300,..;
(224504,523290,913377,1414677,2050345,2847156,3835910), 5051866, 6535206,.;
(5051866,11587072,19918602,30410985,43486800,59633775,79412515,103464895),.;
where the rows are generated as follows.
Start row 0 with all 1's; from then on,
remove the first n terms (shown in parenthesis) from row n
and then take partial sums to yield row n+1.
Note that the main diagonal forms column 0 and equals A101482:
[1,1,2,9,70,795,11961,224504,5051866,132523155,3969912160,...]
which equals column 1 of triangle A101479:
1;
1, 1;
1, 1, 1;
3, 2, 1, 1;
19, 9, 3, 1, 1;
191, 70, 18, 4, 1, 1;
2646, 795, 170, 30, 5, 1, 1;
46737, 11961, 2220, 335, 45, 6, 1, 1;
1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
where row n equals row (n-1) of T^(n-1) with appended '1'.

Crossrefs

Cf. A101479; columns: A101482, A136731, A136732; variants: A136733, A136737.

Programs

  • PARI
    {T(n,k)=if(k<0,0,if(n==0,1,T(n,k-1) + T(n-1,k+n-1)))}

A121427 Main diagonal of rectangular table A121426.

Original entry on oeis.org

1, 2, 15, 178, 2820, 55410, 1294776, 35003430, 1073540871, 36805249870, 1394346324624, 57831360118800, 2605921998840420, 126757491839620950, 6619466939158637640, 369368127676399990338, 21932876159270004129285
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2006

Keywords

Comments

Also main diagonal of square array A136737.

Crossrefs

Cf. A121426, A121412.
Cf. A136737; A121413, A121417, A121422; A136741.

Programs

  • PARI
    {a(n)=local(H=Mat(1), B); for(m=1, n+2, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(H^i)[i-1, j]); )); H=B); return((H^(n+1))[n+2, 2])}

Formula

a(n) = [A121412^(n+1)](n+1,1) for n>=0; i.e., (n+1)-th term of column 1 in matrix power A121412^(n+1).

Extensions

Edited by N. J. A. Sloane, Oct 30 2008 at the suggestion of R. J. Mathar

A136733 Square array, read by antidiagonals, where T(n,k) = T(n,k-1) + T(n-1,k+n) for n>0, k>0, such that T(n,0) = T(n-1,n) for n>0 with T(0,k)=1 for k>=0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 18, 7, 3, 1, 170, 43, 12, 4, 1, 2220, 403, 76, 18, 5, 1, 37149, 5188, 711, 118, 25, 6, 1, 758814, 85569, 9054, 1107, 170, 33, 7, 1, 18301950, 1725291, 147471, 13986, 1605, 233, 42, 8, 1, 508907970, 41145705, 2938176, 225363, 20171, 2220, 308
Offset: 0

Views

Author

Paul D. Hanna, Jan 19 2008

Keywords

Examples

			Square array begins:
(1), 1, 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...];
(1,2), 3, 4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,...;
(3,7,12), 18, 25,33,42,52,63,75,88,102,117,133,150,168,187,207,228,...;
(18,43,76,118), 170, 233,308,396,498,615,748,898,1066,1253,1460,...;
(170,403,711,1107,1605), 2220, 2968,3866,4932,6185,7645,9333,11271,...;
(2220,5188,9054,13986,20171,27816), 37149, 48420,61902,77892,96712,...;
(37149,85569,147471,225363,322075,440785,585046), 758814, 966477,...;
(758814,1725291,2938176,4441557,6285390,8526057,11226958,14459138), ...;
where the rows are generated as follows.
Start row 0 with all 1's; from then on,
remove the first n+1 terms (shown in parenthesis) from row n
and then take partial sums to yield row n+1.
Note the first upper diagonal forms column 0 and equals A101483:
[1,1,3,18,170,2220,37149,758814,18301950,508907970,16023271660,...]
which equals column 2 of triangle A101479:
1;
1, 1;
1, 1, 1;
3, 2, 1, 1;
19, 9, 3, 1, 1;
191, 70, 18, 4, 1, 1;
2646, 795, 170, 30, 5, 1, 1;
46737, 11961, 2220, 335, 45, 6, 1, 1;
1003150, 224504, 37149, 4984, 581, 63, 7, 1, 1; ...
where row n equals row (n-1) of T^(n-1) with appended '1'.

Crossrefs

Cf. A101479; columns: A101483, A121418, A121421; A121425 (main diagonal); variants: A136730, A136737.

Programs

  • PARI
    {T(n,k)=if(k<0,0,if(n==0,1,T(n,k-1) + T(n-1,k+n)))}
Showing 1-6 of 6 results.

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

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