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 1-8 of 8 results.

A121416 Matrix square of triangle A121412.

Original entry on oeis.org

1, 2, 1, 7, 2, 1, 43, 9, 2, 1, 403, 69, 11, 2, 1, 5188, 769, 101, 13, 2, 1, 85569, 11346, 1305, 139, 15, 2, 1, 1725291, 208914, 21745, 2043, 183, 17, 2, 1, 41145705, 4613976, 443329, 37971, 3015, 233, 19, 2, 1, 1133047596, 118840164, 10679494, 850260
Offset: 0

Views

Author

Paul D. Hanna, Aug 22 2006

Keywords

Comments

Row n of triangle T=A121412 equals row (n-1) of T^(n+1) with an appended '1'.

Examples

			Triangle begins:
1;
2, 1;
7, 2, 1;
43, 9, 2, 1;
403, 69, 11, 2, 1;
5188, 769, 101, 13, 2, 1;
85569, 11346, 1305, 139, 15, 2, 1;
1725291, 208914, 21745, 2043, 183, 17, 2, 1;
41145705, 4613976, 443329, 37971, 3015, 233, 19, 2, 1;
1133047596, 118840164, 10679494, 850260, 61860, 4253, 289, 21, 2, 1;

Crossrefs

Cf. A121412, A121420; A101479; columns: A121417, A121418, A121419.

Programs

  • PARI
    {T(n, k)=local(A=Mat(1), B); for(m=1, n+1, 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+1, k+1])}

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

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

A136737 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, 4, 2, 1, 30, 9, 3, 1, 335, 69, 15, 4, 1, 4984, 769, 118, 22, 5, 1, 92652, 11346, 1317, 178, 30, 6, 1, 2065146, 208914, 19311, 1995, 250, 39, 7, 1, 53636520, 4613976, 352636, 29126, 2820, 335, 49, 8, 1, 1589752230, 118840164, 7722840, 528097, 41061
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,...;
(4,9,15,22), 30, 39,49,60,72,85,99,114,130,147,165,184,204,225,247,...;
(30,69,118,178,250), 335, 434,548,678,825,990,1174,1378,1603,1850,...;
(335,769,1317,1995,2820,3810), 4984, 6362,7965,9815,11935,14349,...;
(4984,11346,19311,29126,41061,55410,72492), 92652, 116262, 143722,...;
(92652,208914,352636,528097,740035,993678,1294776,1649634), 2065146,..;
(2065146,4613976,7722840,11476963,15971180,21310710,27611970,35003430,43626510),..;
where the rows are generated as follows.
Start row 0 with all 1's; from then on,
remove the first n+2 terms (shown in parenthesis) from row n
and then take partial sums to yield row n+1.
Note the second upper diagonal forms column 0 and equals A121413:
[1,1,4,30,335,4984,92652,2065146,53636520,1589752230,52926799310,...].
which equals column 3 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: A121413, A121417, A121422; diagonals: A121427, A136741; variants: A136730, A136733.

Programs

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

A121426 Rectangular table, read by antidiagonals, where row n is equal to column 1 of matrix power A121412^(n+1) for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 9, 30, 1, 4, 15, 69, 335, 1, 5, 22, 118, 769, 4984, 1, 6, 30, 178, 1317, 11346, 92652, 1, 7, 39, 250, 1995, 19311, 208914, 2065146, 1, 8, 49, 335, 2820, 29126, 352636, 4613976, 53636520, 1, 9, 60, 434, 3810, 41061, 528097, 7722840, 118840164
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2006

Keywords

Examples

			Table of column 1 in matrix powers of triangle H=A121412 begins:
H^1: 1, 1, 4, 30, 335, 4984, 92652, 2065146, 53636520, ...
H^2: 1, 2, 9, 69, 769, 11346, 208914, 4613976, 118840164, ...
H^3: 1, 3, 15, 118, 1317, 19311, 352636, 7722840, 197354133, ...
H^4: 1, 4, 22, 178, 1995, 29126, 528097, 11476963, 291124693, ...
H^5: 1, 5, 30, 250, 2820, 41061, 740035, 15971180, 402319275, ...
H^6: 1, 6, 39, 335, 3810, 55410, 993678, 21310710, 533345745, ...
H^7: 1, 7, 49, 434, 4984, 72492, 1294776, 27611970, 686872893, ...
H^8: 1, 8, 60, 548, 6362, 92652, 1649634, 35003430, 865852191, ...
H^9: 1, 9, 72, 678, 7965, 116262, 2065146, 43626510, 1073540871, ...
Rearrangement of the upper part of the table forms A121431, which is
the number of subpartitions of partition [0,0,1,1,1,2,2,2,2,3,...]:
1,1, 1,2,3, 4,9,15,22, 30,69,118,178,250, 335,769,1317,1995,2820,...

Crossrefs

Cf. A121427 (diagonal), A121431; rows: A121413, A121417, A121422; related tables: A121424, A121428; related triangles: A121412, A121416, A121420.

Programs

  • PARI
    {T(n,k)=local(H=Mat(1), B); for(m=1, k+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))[k+2, 2])}

A121430 Number of subpartitions of partition P=[0,1,1,2,2,2,3,3,3,3,4,...] (A003056).

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 18, 43, 76, 118, 170, 403, 711, 1107, 1605, 2220, 5188, 9054, 13986, 20171, 27816, 37149, 85569, 147471, 225363, 322075, 440785, 585046, 758814, 1725291, 2938176, 4441557, 6285390, 8526057, 11226958, 14459138, 18301950
Offset: 0

Views

Author

Paul D. Hanna, Jul 30 2006

Keywords

Comments

See A115728 for the definition of subpartitions of a partition.

Examples

			The g.f. is illustrated by:
1 = (1)*(1-x)^1 + (x + 2*x^2)*(1-x)^2 +
(3*x^3 + 7*x^5 + 12*x^6)*(1-x)^3 +
(18*x^6 + 43*x^7 + 76*x^8 + 118*x^9)*(1-x)^4 +
(170*x^10 + 403*x^11 + 711*x^12 + 1107*x^13 + 1605*x^14)*(1-x)^5 + ...
When the sequence is put in the form of a triangle:
1;
1, 2;
3, 7, 12;
18, 43, 76, 118;
170, 403, 711, 1107, 1605;
2220, 5188, 9054, 13986, 20171, 27816;
37149, 85569, 147471, 225363, 322075, 440785, 585046; ...
then the columns of this triangle form column 0 (with offset)
of successive matrix powers of triangle H=A121412.
This sequence is embedded in table A121424 as follows.
Column 0 of successive powers of matrix H begin:
H^1: [1,1,3,18,170,2220,37149,758814,18301950,...];
H^2: 1, [2,7,43,403,5188,85569,1725291,41145705,...];
H^3: 1,3, [12,76,711,9054,147471,2938176,69328365,...];
H^4: 1,4,18, [118,1107,13986,225363,4441557,103755660,...];
H^5: 1,5,25,170, [1605,20171,322075,6285390,145453290,...];
H^6: 1,6,33,233,2220, [27816,440785,8526057,195579123,...];
H^7: 1,7,42,308,2968,37149, [585046,11226958,255436293,...];
H^8: 1,8,52,396,3866,48420,758814, [14459138,326487241,...];
H^9: 1,9,63,498,4932,61902,966477,18301950, [410368743,...];
the terms enclosed in brackets form this sequence.

Crossrefs

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121424, A121425; column 0 of H^n: A121413, A121417, A121421.

Programs

  • PARI
    {a(n)=local(A); if(n==0,1,A=x+x*O(x^n); for(k=0, n, A+=polcoeff(A, k)*x^k*(1-(1-x)^( (sqrtint(8*k+1)+1)\2 ) )); polcoeff(A, n))}

Formula

G.f.: 1 = Sum_{n>=1} (1-x)^n * Sum_{k=n*(n-1)/2..n*(n+1)/2-1} a(k)*x^k.

A121419 Column 2 of triangle A121416.

Original entry on oeis.org

1, 2, 11, 101, 1305, 21745, 443329, 10679494, 296547736, 9319259500, 326788327650, 12643827604842, 534889691765631, 24555735428777265, 1215611513578215355, 64542477563559758310, 3658333757447085090365
Offset: 0

Views

Author

Paul D. Hanna, Aug 22 2006

Keywords

Comments

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: A121417, A121418.

Programs

  • PARI
    {a(n)=local(A=Mat(1), B); for(m=1, n+3, 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+3, 3])}

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.
Showing 1-8 of 8 results.

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

Content is available under The OEIS End-User License Agreement.