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 12 results. Next

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

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

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])}

A121412 Triangular matrix T, read by rows, where row n of T equals row (n-1) of T^(n+1) with an appended '1'.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 18, 4, 1, 1, 170, 30, 5, 1, 1, 2220, 335, 45, 6, 1, 1, 37149, 4984, 581, 63, 7, 1, 1, 758814, 92652, 9730, 924, 84, 8, 1, 1, 18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1, 508907970, 53636520, 4843125, 387567, 28365, 1965, 135, 10, 1, 1
Offset: 0

Views

Author

Paul D. Hanna, Jul 30 2006

Keywords

Comments

Related to the number of subpartitions of a partition as defined in A115728; for examples involving column k of successive matrix powers, see A121430, A121431, A121432 and A121433. Essentially the same as triangle A101479, but this form best illustrates the nice properties of this triangle.

Examples

			Triangle T begins:
1;
1, 1;
3, 1, 1;
18, 4, 1, 1;
170, 30, 5, 1, 1;
2220, 335, 45, 6, 1, 1;
37149, 4984, 581, 63, 7, 1, 1;
758814, 92652, 9730, 924, 84, 8, 1, 1;
18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1;
508907970, 53636520, 4843125, 387567, 28365, 1965, 135, 10, 1, 1;
To get row 4 of T, append '1' to row 3 of matrix power T^5:
1;
5, 1;
25, 5, 1;
170, 30, 5, 1; ...
To get row 5 of T, append '1' to row 4 of matrix power T^6:
1;
6, 1;
33, 6, 1;
233, 39, 6, 1;
2220, 335, 45, 6, 1; ...
Likewise, get row n of T by appending '1' to row (n-1) of T^(n+1).

Links

Crossrefs

Cf. A121416 (T^2), A121420 (T^3), columns: A121413, A121414, A121415; related tables: A121424, A121426, A121428; related subpartitions: A121430, A121431, A121432, A121433.

Programs

  • Mathematica
    T[n_, k_] := Module[{A = {{1}}, B}, Do[B = Array[0&, {m, m}]; Do[Do[B[[i, j]] = If[j == i, 1, MatrixPower[A, i][[i-1, j]]], {j, 1, i}], {i, 1, m}]; A = B, {m, 1, n+1}]; A[[n+1, k+1]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 03 2019 *)
  • PARI
    {T(n, k) = my(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^1)[n+1, k+1])}
for(n=0,12, for(k=0,n, print1( T(n,k),", "));print(""))

Formula

G.f.: Column k of successive powers of T satisfy the amazing relation given by: 1 = Sum_{n>=0} (1-x)^(n+1) * x^(n(n+1)/2 + k*n) * Sum_{j=0..n+k} [T^(j+1)](n+k,k) * x^j.

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])}

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])}

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.

A121431 Number of subpartitions of partition P=[0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,...] (A052146).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 9, 15, 22, 30, 69, 118, 178, 250, 335, 769, 1317, 1995, 2820, 3810, 4984, 11346, 19311, 29126, 41061, 55410, 72492, 92652, 208914, 352636, 528097, 740035, 993678, 1294776, 1649634, 2065146, 4613976, 7722840, 11476963, 15971180
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. may be illustrated by:
1/(1-x) = (1 + 1*x)*(1-x)^0 + (x^2 + 2*x^3 + 3*x^4)*(1-x)^1 +
(4*x^5 + 9*x^6 + 15*x^7 + 22*x^8)*(1-x)^2 +
(30*x^9 + 69*x^10 + 118*x^11 + 178*x^12 + 250*x^13)*(1-x)^3 +
(335*x^14 + 769*x^15 + 1317*x^16 + 1995*x^17 + 2820*x^18 + 3810*x^19)*(1-x)^4 +...
When the sequence is put in the form of a triangle:
1, 1,
1, 2, 3,
4, 9, 15, 22,
30, 69, 118, 178, 250,
335, 769, 1317, 1995, 2820, 3810,
4984, 11346, 19311, 29126, 41061, 55410, 72492,
92652, 208914, 352636, 528097, 740035, 993678, 1294776, ...
then the columns of this triangle form column 1 (with offset)
of successive matrix powers of triangle H=A121412.
This sequence is embedded in table A121426 as follows.
Column 1 of successive powers of matrix H begin:
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,...];
the terms enclosed in brackets form this sequence.

Crossrefs

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121426, A121427; column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121432, A121433.

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+9)+1)\2 - 1 ) )); polcoeff(A, n))}

Formula

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^A052146(n).
Showing 1-10 of 12 results. Next

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