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 1-10 of 29 results. Next

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])}

A121420 Matrix cube of triangle A121412.

Original entry on oeis.org

1, 3, 1, 12, 3, 1, 76, 15, 3, 1, 711, 118, 18, 3, 1, 9054, 1317, 169, 21, 3, 1, 147471, 19311, 2190, 229, 24, 3, 1, 2938176, 352636, 36360, 3378, 298, 27, 3, 1, 69328365, 7722840, 737051, 62655, 4929, 376, 30, 3, 1, 1891371807, 197354133, 17645187
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 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;
3, 1;
12, 3, 1;
76, 15, 3, 1;
711, 118, 18, 3, 1;
9054, 1317, 169, 21, 3, 1;
147471, 19311, 2190, 229, 24, 3, 1;
2938176, 352636, 36360, 3378, 298, 27, 3, 1;
69328365, 7722840, 737051, 62655, 4929, 376, 30, 3, 1; ...

Crossrefs

Cf. A121412, A121416; A101479; columns: A121421, A121422, A121423.

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^3)[n+1, k+1])}

A121413 Column 1 of triangle A121412, in which row n of T equals row (n-1) of T^(n+1) with an appended '1'.

Original entry on oeis.org

1, 1, 4, 30, 335, 4984, 92652, 2065146, 53636520, 1589752230, 52926799310, 1954793723688, 79297921097850, 3504206787532290, 167533486104374420, 8615231103973721610, 474145710046504872894, 27806642356199323970739
Offset: 0

Views

Author

Paul D. Hanna, Aug 22 2006

Keywords

Comments

Also equals column 3 of triangle A101479.

Crossrefs

Cf. A121412 (triangle); other columns: A101483, A121414, A121415; A101479 (triangle).

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^1)[n+2, 2])}

A121414 Column 2 of triangle A121412, in which row n of T=A121412 equals row (n-1) of T^(n+1) with an appended '1'.

Original entry on oeis.org

1, 1, 5, 45, 581, 9730, 199692, 4843125, 135345925, 4278317373, 150818840250, 5863215069621, 249105031449435, 11480173020040450, 570303168053225908, 30375972794764190385, 1726700205634807475115, 104332294453480284687895
Offset: 0

Views

Author

Paul D. Hanna, Aug 22 2006

Keywords

Comments

Also equals column 4 of triangle A101479.

Crossrefs

Cf. A121412 (triangle); other columns: A101483, A121413, A121415; A101479 (triangle).

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^1)[n+3, 3])}

A121424 Rectangular table, read by antidiagonals, where row n is equal to column 0 of matrix power A121412^(n+1) for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 18, 1, 4, 12, 43, 170, 1, 5, 18, 76, 403, 2220, 1, 6, 25, 118, 711, 5188, 37149, 1, 7, 33, 170, 1107, 9054, 85569, 758814, 1, 8, 42, 233, 1605, 13986, 147471, 1725291, 18301950, 1, 9, 52, 308, 2220, 20171, 225363, 2938176, 41145705
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2006

Keywords

Examples

			Table of column 0 in matrix powers of triangle H=A121412 begins:
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,...
Rearrangement of the upper half of the table forms A121430, which is
the number of subpartitions of partition [0,1,1,2,2,2,3,3,3,3,4,...]:
1, 1,2, 3,7,12, 18,43,76,118, 170,403,711,1107,1605, 2220,...

Crossrefs

Cf. A121425 (diagonal), A121430; rows: A101483, A121418, A121421; related tables: A121426, A121428; related triangles: A121412, A121416, A121420.

Programs

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

A121428 Rectangular table, read by antidiagonals, where row n is equal to column 2 of matrix power A121412^(n+1) for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 1, 3, 11, 45, 1, 4, 18, 101, 581, 1, 5, 26, 169, 1305, 9730, 1, 6, 35, 250, 2190, 21745, 199692, 1, 7, 45, 345, 3255, 36360, 443329, 4843125, 1, 8, 56, 455, 4520, 53916, 737051, 10679494, 135345925, 1, 9, 68, 581, 6006, 74781, 1087583, 17645187
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2006

Keywords

Examples

			Table of column 2 in matrix powers of triangle H=A121412 begins:
H^1: 1, 1, 5, 45, 581, 9730, 199692, 4843125, 135345925, ...
H^2: 1, 2, 11, 101, 1305, 21745, 443329, 10679494, 296547736, ...
H^3: 1, 3, 18, 169, 2190, 36360, 737051, 17645187, 487025244, ...
H^4: 1, 4, 26, 250, 3255, 53916, 1087583, 25889969, 710546530, ...
H^5: 1, 5, 35, 345, 4520, 74781, 1502270, 35578270, 971255050, ...
H^6: 1, 6, 45, 455, 6006, 99351, 1989113, 46890210, 1273698270, ...
H^7: 1, 7, 56, 581, 7735, 128051, 2556806, 60022670, 1622857887, ...
H^8: 1, 8, 68, 724, 9730, 161336, 3214774, 75190410, 2024181693, ...
H^9: 1, 9, 81, 885, 12015, 199692, 3973212, 92627235, 2483617140, ...
Rearrangement of the upper part of the table forms A121432, which is
the number of subpartitions of partition [0,0,0,1,1,1,1,2,2,2,2,2,..]:
1,1,1, 1,2,3,4, 5,11,18,26,35, 45,101,169,250,345,455, 581,1305,...

Crossrefs

Cf. A121429 (diagonal), A121431; rows: A121414, A121419, A121423; related tables: A121424, A121426; related triangles: A121412, A121416, A121420.

Programs

  • PARI
    {T(n,k)=local(H=Mat(1), B); for(m=1, k+3, 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+3, 3])}

A121415 Column 3 of triangle A121412, in which row n of T=A121412 equals row (n-1) of T^(n+1) with an appended '1'.

Original entry on oeis.org

1, 1, 6, 63, 924, 17226, 387567, 10182744, 305379129, 10280074116, 383492465902, 15692864353299, 698622377024472, 33604795914668178, 1736477536255603281, 95918139377302294980, 5639487915973132301793, 351611645300506492405623
Offset: 0

Views

Author

Paul D. Hanna, Aug 22 2006

Keywords

Comments

Also equals column 5 of triangle A101479.

Crossrefs

Cf. A121412 (triangle); other columns: A101483, A121413, A121414; A101479 (triangle).

Programs

  • PARI
    {a(n)=local(A=Mat(1), B); for(m=1, n+4, 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^1)[n+4, 4])}

A121418 Column 0 of triangle A121416.

Original entry on oeis.org

1, 2, 7, 43, 403, 5188, 85569, 1725291, 41145705, 1133047596, 35377360292, 1234796503280, 47636225803285, 2012509471127885, 92398547122062997, 4580472438441602301, 243822925502110419105, 13870297863425823346284
Offset: 0

Views

Author

Paul D. Hanna, Aug 22 2006, Jan 19 2008

Keywords

Comments

Also column 1 of square array A136733.
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. A136733; A101483, A121421; A121425.

Programs

  • PARI
    {a(n)=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, 1])}

Extensions

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

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
Showing 1-10 of 29 results. Next

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

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