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 1-5 of 5 results.

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.

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])}

A121432 Number of subpartitions of partition P=[0,0,0,1,1,1,1,2,2,2,2,2,3,...], where P(n) = [(sqrt(8*n+25) - 5)/2].

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 11, 18, 26, 35, 45, 101, 169, 250, 345, 455, 581, 1305, 2190, 3255, 4520, 6006, 7735, 9730, 21745, 36360, 53916, 74781, 99351, 128051, 161336, 199692, 443329, 737051, 1087583, 1502270, 1989113, 2556806, 3214774, 3973212, 4843125
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 + x + x^2)*(1-x)^0 + (x^3 + 2*x^4 + 3*x^5 + 4*x^6)*(1-x)^1 +
(5*x^7 + 11*x^8 + 18*x^9 + 26*x^10 + 35*x^11)*(1-x)^2 +
(45*x^12 + 101*x^13 + 169*x^14 + 250*x^15 + 345*x^16 + 455*x^17)*(1-x)^3 +
(581*x^18 + 1305*x^19 + 2190*x^20 + 3255*x^21 + 4520*x^22 + 6006*x^23 + 7735*x^24)*(1-x)^4 +...
When the sequence is put in the form of a triangle:
1, 1, 1,
1, 2, 3, 4,
5, 11, 18, 26, 35,
45, 101, 169, 250, 345, 455,
581, 1305, 2190, 3255, 4520, 6006, 7735,
9730, 21745, 36360, 53916, 74781, 99351, 128051, 161336, ...
then the columns of this triangle form column 2 (with offset)
of successive matrix powers of triangle H=A121412.
This sequence is embedded in table A121428 as follows.
Column 2 of successive powers of matrix H begin:
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,...];
the terms enclosed in brackets form this sequence.

Crossrefs

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121428, A121429; column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121431, 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+25)+1)\2 - 2 ) )); polcoeff(A, n))}

Formula

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^P(n), where P(n)=[(sqrt(8*n+25)-5)/2].

A121433 Number of subpartitions of partition P=[0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,...], where P(n) = [(sqrt(8*n+49) - 7)/2].

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 13, 21, 30, 40, 51, 63, 139, 229, 334, 455, 593, 749, 924, 2043, 3378, 4951, 6785, 8904, 11333, 14098, 17226, 37971, 62655, 91728, 125671, 164997, 210252, 262016, 320904, 387567, 850260, 1397268, 2038545, 2784850, 3647788
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 + x + x^2 + x^3)*(1-x)^0 +
(x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8)*(1-x)^1 +
(6*x^9 + 13*x^10 + 21*x^11 + 30*x^12 + 40*x^13 + 51*x^14)*(1-x)^2 +
(63*x^15 + 139*x^16 + 229*x^17 + 334*x^18 + 455*x^19 + 593*x^20 + 749*x^21)*(1-x)^3 +
When the sequence is put in the form of a triangle:
1, 1, 1, 1,
1, 2, 3, 4, 5,
6, 13, 21, 30, 40, 51,
63, 139, 229, 334, 455, 593, 749,
924, 2043, 3378, 4951, 6785, 8904, 11333, 14098,
17226, 37971, 62655, 91728, 125671, 164997, 210252, 262016, 320904,
then the columns of this triangle form column 3 (with offset)
of successive matrix powers of triangle H=A121412.
Column 3 of successive powers of matrix H begin:
H^1: [1,1,6,63,924,17226,387567,10182744,305379129,...];
H^2: [1,2,13,139,2043,37971,850260,22224723,663173878,...];
H^3: [1,3,21,229,3378,62655,1397268,36351147,1079567193,...];
H^4: [1,4,30,334,4951,91728,2038545,52807195,1561301733,...];
H^5: 1, [5,40,455,6785,125671,2784850,71859275,2115718545,...];
H^6: 1,6, [51,593,8904,164997,3647788,93796335,2750797677,...];
H^7: 1,7,63, [749,11333,210252,4639852,118931226,3475200792,...];
H^8: 1,8,76,924, [14098,262016,5774466,147602118,4298315847,...];
H^9: 1,9,90,1119,17226, [320904,7066029,180173970,5230303902,...];
the terms enclosed in brackets form this sequence.

Crossrefs

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121431, A121432.

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+49)+1)\2 - 3 ) )); polcoeff(A, n))}

Formula

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^P(n), where P(n)=[(sqrt(8*n+49)-7)/2].
Showing 1-5 of 5 results.

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