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.

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])}

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

A121425 Main diagonal of rectangular table A121424.

Original entry on oeis.org

1, 2, 12, 118, 1605, 27816, 585046, 14459138, 410368743, 13146830110, 469123986529, 18447791712945, 792514583941223, 36925394368325295, 1854525584914459755, 99872579714406393286, 5740977285851988017769
Offset: 0

Views

Author

Paul D. Hanna, Aug 26 2006

Keywords

Comments

Also main diagonal of square array A136733.

Crossrefs

Cf. A121424, A121412.
Cf. A136733; A101483, A121418, A121421.

Programs

  • PARI
    {a(n)=local(H=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]=(H^i)[i-1, j]); )); H=B); return((H^(n+1))[n+1, 1])}

Formula

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

Extensions

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

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])}

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.

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)))}

A121423 Column 2 of triangle A121420.

Original entry on oeis.org

1, 3, 18, 169, 2190, 36360, 737051, 17645187, 487025244, 15219471545, 530951735025, 20447695079658, 861389893507518, 39394187817328680, 1943446826192453505, 102863050524539640435, 5813722327999905078450
Offset: 0

Views

Author

Paul D. Hanna, Aug 23 2006

Keywords

Comments

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, A121422.

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^3)[n+3, 3])}
Showing 1-8 of 8 results.

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

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