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.

Previous Showing 21-25 of 25 results.

A193602 Augmentation of the triangle A122366. See Comments.

Original entry on oeis.org

1, 1, 3, 1, 8, 19, 1, 15, 80, 172, 1, 24, 221, 971, 1967, 1, 35, 492, 3547, 13809, 26832, 1, 48, 955, 10186, 62840, 224529, 422609, 1, 63, 1684, 24890, 222132, 1226003, 4102449, 7525966, 1, 80, 2765, 54077, 658319, 5167948, 26193697, 83159133
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Examples

			First five rows of A193602:
1
1...3
1...8....19
1...15...80...172
1...24...221..971...1967

Crossrefs

Cf. A122366, A193091.

Programs

  • Mathematica
    p[n_, k_] := Binomial[2 n + 1, k] (* A122366 *)
Table[p[n, k], {n, 0, 7}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193602 *)
Flatten[Table[v[n], {n, 0, 8}]]

A193603 Augmentation of the triangle A008949. See Comments.

Original entry on oeis.org

1, 1, 2, 1, 5, 8, 1, 9, 30, 44, 1, 14, 77, 212, 296, 1, 20, 163, 700, 1712, 2312, 1, 27, 305, 1877, 6882, 15476, 20384, 1, 35, 523, 4365, 22380, 73240, 154424, 199376, 1, 44, 840, 9134, 62479, 280630, 841312, 1683992, 2138336
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091. The right edge of the triangle A193603 is A111537.

Examples

			First five rows of A193603:
1
1...2
1...5....8
1...9...30....44
1...14...77..212...296

Crossrefs

Cf. A008949, A193091, A111537.

Programs

  • Mathematica
    p[n_, k_] := Sum[Binomial[n, h], {h, 0, k}] (* A008949 *)
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193603 *)
Flatten[Table[v[n], {n, 0, 8}]]

A193604 Augmentation of the triangle A055248. See Comments.

Original entry on oeis.org

1, 2, 1, 8, 8, 3, 64, 88, 62, 19, 1024, 1664, 1568, 896, 233, 32768, 58368, 64128, 49248, 23890, 5385, 2097152, 3932160, 4703232, 4249728, 2800488, 1179656, 233787, 268435456, 517996544, 649887744, 645547008, 507802304, 293645688, 108978862
Offset: 0

Views

Author

Clark Kimberling, Jul 31 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.

Examples

			First five rows of A193604:
1
2.....1
8.....8.....3
64....88....62....19
1024..1664..1568..896...233

Crossrefs

Cf. A055248, A193091.

Programs

  • Mathematica
    u[n_, k_] := Sum[Binomial[n, h], {h, 0, k}] (* A055248 *)
p[n_, k_] := u[n, n - k]
Table[p[n, k], {n, 0, 5}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193604 *)
Flatten[Table[v[n], {n, 0, 8}]]

A193630 Augmentation of the triangle A074909. See Comments.

Original entry on oeis.org

1, 1, 2, 1, 5, 7, 1, 9, 28, 33, 1, 14, 74, 181, 191, 1, 20, 159, 637, 1333, 1297, 1, 27, 300, 1767, 5906, 11029, 10063, 1, 35, 517, 4190, 20256, 59324, 101351, 87669, 1, 44, 833, 8873, 58339, 244125, 645146, 1024949, 847015, 1, 54, 1274, 17241, 147680
Offset: 0

Views

Author

Clark Kimberling, Aug 01 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
Regarding A193630, writing the general term as w(n,k),
w(n,n): A104981
w(n,n-1): A156629

Examples

			First five rows of A193607:
1
1...2
1...5....7
1...9....28...33
1...14...74...181...191

Crossrefs

Cf. A193091, A074909.

Programs

  • Mathematica
    p[n_, k_] := Binomial[n + 1, k];
Table[p[n, k], {n, 0, 7}, {k, 0, n}]  (* A074909 *)
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193630 *)
Flatten[Table[v[n], {n, 0, 8}]]

A193631 Augmentation of the triangle given by p(n,k)=(3+(-1)^k)/2 for 0<=k<=n. See Comments.

Original entry on oeis.org

1, 2, 1, 4, 4, 5, 8, 12, 22, 17, 16, 32, 72, 88, 89, 32, 80, 208, 328, 474, 417, 64, 192, 560, 1056, 1836, 2364, 2253, 128, 448, 1440, 3120, 6168, 9684, 13038, 11937, 256, 1024, 3584, 8704, 19040, 34240, 54800, 71152, 66737, 512, 2304, 8704, 23296
Offset: 0

Views

Author

Clark Kimberling, Aug 01 2011

Keywords

Comments

For an introduction to the unary operation "augmentation" as applied to triangular arrays or sequences of polynomials, see A193091.
(column 2 of A193631)=A001787.

Examples

			First five rows of the triangle P=p(n,k):
2
2...1
2...1...2
2...1...2...1
2...1...2...1...2
First five rows of A193631:
1
2....1
4....4....5
8....12...22...17
16...32...72...88...89

Crossrefs

Cf. A193091, A001787.

Programs

  • Mathematica
    p[n_, k_] := (3 + (-1)^k)/2;
Table[p[n, k], {n, 0, 7}, {k, 0, n}]
m[n_] := Table[If[i <= j, p[n + 1 - i, j - i], 0], {i, n}, {j, n + 1}]
TableForm[m[4]]
w[0, 0] = 1; w[1, 0] = p[1, 0]; w[1, 1] = p[1, 1];
v[0] = w[0, 0]; v[1] = {w[1, 0], w[1, 1]};
v[n_] := v[n - 1].m[n]
TableForm[Table[v[n], {n, 0, 6}]] (* A193631 *)
Flatten[Table[v[n], {n, 0, 8}]]
Previous Showing 21-25 of 25 results.

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