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 11-16 of 16 results.

A157177 A new general triangle sequence based on the Eulerian form in three parts:m=1; t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]] t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) + (m*k + 1)*t0(n - 1 + 1, k) + m*k*(n - k)*t0(n - 2 + 1, k - 1)].

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 82, 29, 1, 1, 61, 368, 368, 61, 1, 1, 125, 1399, 3010, 1399, 125, 1, 1, 253, 4863, 19243, 19243, 4863, 253, 1, 1, 509, 16048, 106099, 194846, 106099, 16048, 509, 1, 1, 1021, 51298, 532466, 1622734, 1622734, 532466, 51298
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Feb 24 2009

Keywords

Comments

Row sums are:
{1, 2, 7, 28, 142, 860, 6060, 48720, 440160, 4415040, 48686400,...}.
The m=1 of the general sequence is A008518.

Examples

			{1},
{1, 1},
{1, 5, 1},
{1, 13, 13, 1},
{1, 29, 82, 29, 1},
{1, 61, 368, 368, 61, 1},
{1, 125, 1399, 3010, 1399, 125, 1},
{1, 253, 4863, 19243, 19243, 4863, 253, 1},
{1, 509, 16048, 106099, 194846, 106099, 16048, 509, 1},
{1, 1021, 51298, 532466, 1622734, 1622734, 532466, 51298, 1021, 1},
{1, 2045, 160669, 2510256, 11855730, 19628998, 11855730, 2510256, 160669, 2045, 1}

Crossrefs

A008518

Programs

  • Mathematica
    Clear[t, n, k, m];
t[n_, k_, m_] = (m*(n - k) + 1)*Binomial[n - 1, k - 1] + (m*k + 1)*Binomial[n - 1, k] - m*k*(n - k)*Binomial[n - 2, k - 1];
Table[t[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
Table[Flatten[Table[Table[t[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
Table[Table[Sum[t[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

Formula

m=1;
t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]];
t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) +
(m*k + 1)*t0(n - 1 + 1, k) +
m*k*(n - k)*t0(n - 2 + 1, k - 1)].

A157178 A new general triangle sequence based on the Eulerian form in three parts:m=2; t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]] t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) + (m*k + 1)*t0(n - 1 + 1, k) + m*k*(n - k)*t0(n - 2 + 1, k - 1)].

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 21, 21, 1, 1, 46, 142, 46, 1, 1, 95, 644, 644, 95, 1, 1, 192, 2439, 5416, 2439, 192, 1, 1, 385, 8415, 34879, 34879, 8415, 385, 1, 1, 770, 27556, 192286, 358454, 192286, 27556, 770, 1, 1, 1539, 87486, 962090, 3001044, 3001044, 962090, 87486
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Feb 24 2009

Keywords

Comments

Row sums are:
{1, 2, 10, 44, 236, 1480, 10680, 87360, 799680, 8104320, 90115200,...}.
The m=1 of the general sequence is A008518.

Examples

			{1},
{1, 1},
{1, 8, 1},
{1, 21, 21, 1},
{1, 46, 142, 46, 1},
{1, 95, 644, 644, 95, 1},
{1, 192, 2439, 5416, 2439, 192, 1},
{1, 385, 8415, 34879, 34879, 8415, 385, 1},
{1, 770, 27556, 192286, 358454, 192286, 27556, 770, 1},
{1, 1539, 87486, 962090, 3001044, 3001044, 962090, 87486, 1539, 1},
{1, 3076, 272485, 4517480, 21945914, 36637288, 21945914, 4517480, 272485, 3076, 1}

Crossrefs

A008518

Programs

  • Mathematica
    Clear[t, n, k, m];
t[n_, k_, m_] = (m*(n - k) + 1)*Binomial[n - 1, k - 1] + (m*k + 1)*Binomial[n - 1, k] - m*k*(n - k)*Binomial[n - 2, k - 1];
Table[t[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
Table[Flatten[Table[Table[t[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
Table[Table[Sum[t[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

Formula

m=2;
t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]];
t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) +
(m*k + 1)*t0(n - 1 + 1, k) +
m*k*(n - k)*t0(n - 2 + 1, k - 1)].

A157179 A new general triangle sequence based on the Eulerian form in three parts ( subtraction):m=1; t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]] t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) + (m*k + 1)*t0(n - 1 + 1, k) - m*k*(n - k)*t0(n - 2 + 1, k - 1)].

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 9, 1, 1, 23, 50, 23, 1, 1, 53, 236, 236, 53, 1, 1, 115, 983, 1822, 983, 115, 1, 1, 241, 3723, 11995, 11995, 3723, 241, 1, 1, 495, 13168, 70369, 117534, 70369, 13168, 495, 1, 1, 1005, 44382, 377918, 997974, 997974, 377918, 44382, 1005, 1, 1
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Feb 24 2009

Keywords

Comments

Row sums are:
{1, 2, 5, 20, 98, 580, 4020, 31920, 285600, 2842560, 31147200,...}.
The m=0 of the general sequence is A008518.

Examples

			{1},
{1, 1},
{1, 3, 1},
{1, 9, 9, 1},
{1, 23, 50, 23, 1},
{1, 53, 236, 236, 53, 1},
{1, 115, 983, 1822, 983, 115, 1},
{1, 241, 3723, 11995, 11995, 3723, 241, 1},
{1, 495, 13168, 70369, 117534, 70369, 13168, 495, 1},
{1, 1005, 44382, 377918, 997974, 997974, 377918, 44382, 1005, 1},
{1, 2027, 144605, 1896720, 7620498, 11819498, 7620498, 1896720, 144605, 2027, 1}

Crossrefs

A008518

Programs

  • Mathematica
    Clear[t, n, k, m];
t[n_, k_, m_] = (m*(n - k) + 1)*Binomial[n - 1, k - 1] + (m*k + 1)*Binomial[n - 1, k] - m*k*(n - k)*Binomial[n - 2, k - 1];
Table[t[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
Table[Flatten[Table[Table[t[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
Table[Table[Sum[t[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

Formula

m=1;
t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]];
t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) +
(m*k + 1)*t0(n - 1 + 1, k) +
m*k*(n - k)*t0(n - 2 + 1, k - 1)].

A157180 A new general triangle sequence based on the Eulerian form in three parts ( subtraction):m=2; t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]] t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) + (m*k + 1)*t0(n - 1 + 1, k) - m*k*(n - k)*t0(n - 2 + 1, k - 1)].

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 34, 78, 34, 1, 1, 79, 380, 380, 79, 1, 1, 172, 1607, 3040, 1607, 172, 1, 1, 361, 6135, 20383, 20383, 6135, 361, 1, 1, 742, 21796, 120826, 203830, 120826, 21796, 742, 1, 1, 1507, 73654, 652994, 1751524, 1751524, 652994, 73654
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Feb 24 2009

Keywords

Comments

Row sums are:
{1, 2, 6, 28, 148, 920, 6600, 53760, 490560, 4959360, 55036800,...}.
The m=0 of the general sequence is A008518.

Examples

			{1},
{1, 1},
{1, 4, 1},
{1, 13, 13, 1},
{1, 34, 78, 34, 1},
{1, 79, 380, 380, 79, 1},
{1, 172, 1607, 3040, 1607, 172, 1},
{1, 361, 6135, 20383, 20383, 6135, 361, 1},
{1, 742, 21796, 120826, 203830, 120826, 21796, 742, 1},
{1, 1507, 73654, 652994, 1751524, 1751524, 652994, 73654, 1507, 1},
{1, 3040, 240357, 3290408, 13475450, 21018288, 13475450, 3290408, 240357, 3040, 1}

Crossrefs

A008518

Programs

  • Mathematica
    Clear[t, n, k, m];
t[n_, k_, m_] = (m*(n - k) + 1)*Binomial[n - 1, k - 1] + (m*k + 1)*Binomial[n - 1, k] - m*k*(n - k)*Binomial[n - 2, k - 1];
Table[t[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
Table[Flatten[Table[Table[t[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
Table[Table[Sum[t[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

Formula

m=2;
t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]];
t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) +
(m*k + 1)*t0(n - 1 + 1, k) +
m*k*(n - k)*t0(n - 2 + 1, k - 1)].

A157181 A new general triangle sequence based on the Eulerian form in three parts ( subtraction):m=3; t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]] t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) + (m*k + 1)*t0(n - 1 + 1, k) - m*k*(n - k)*t0(n - 2 + 1, k - 1)].

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 17, 17, 1, 1, 45, 106, 45, 1, 1, 105, 524, 524, 105, 1, 1, 229, 2231, 4258, 2231, 229, 1, 1, 481, 8547, 28771, 28771, 8547, 481, 1, 1, 989, 30424, 171283, 290126, 171283, 30424, 989, 1, 1, 2009, 102926, 928070, 2505074, 2505074, 928070
Offset: 0

Views

Author

Roger L. Bagula and Gary W. Adamson, Feb 24 2009

Keywords

Comments

Row sums are:
{1, 2, 7, 36, 198, 1260, 9180, 75600, 695520, 7076160, 78926400,...}.
The m=0 of the general sequence is A008518.

Examples

			{1},
{1, 1},
{1, 5, 1},
{1, 17, 17, 1},
{1, 45, 106, 45, 1},
{1, 105, 524, 524, 105, 1},
{1, 229, 2231, 4258, 2231, 229, 1},
{1, 481, 8547, 28771, 28771, 8547, 481, 1},
{1, 989, 30424, 171283, 290126, 171283, 30424, 989, 1},
{1, 2009, 102926, 928070, 2505074, 2505074, 928070, 102926, 2009, 1},
{1, 4053, 336109, 4684096, 19330402, 30217078, 19330402, 4684096, 336109, 4053, 1}

Crossrefs

A008518

Programs

  • Mathematica
    Clear[t, n, k, m];
t[n_, k_, m_] = (m*(n - k) + 1)*Binomial[n - 1, k - 1] + (m*k + 1)*Binomial[n - 1, k] - m*k*(n - k)*Binomial[n - 2, k - 1];
Table[t[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
Table[Flatten[Table[Table[t[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
Table[Table[Sum[t[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

Formula

m=3;
t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]];
t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) +
(m*k + 1)*t0(n - 1 + 1, k) +
m*k*(n - k)*t0(n - 2 + 1, k - 1)].

A177984 A symmetrical triangle of polynomial coefficients:p(x,n)=If[n == 0, 1, (1 - x)^(n + 1)*Sum[((2*k + 1)^n + (k + 1)^n + k^n)*x^k, {k, 0, Infinity}]/2].

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 14, 14, 1, 1, 44, 126, 44, 1, 1, 132, 887, 887, 132, 1, 1, 390, 5451, 12076, 5451, 390, 1, 1, 1150, 30984, 131665, 131665, 30984, 1150, 1, 1, 3400, 168076, 1252600, 2353126, 1252600, 168076, 3400, 1, 1, 10088, 885725, 10905407, 34828859
Offset: 0

Views

Author

Roger L. Bagula, May 16 2010

Keywords

Comments

Row sums are:
{1, 2, 6, 30, 216, 2040, 23760, 327600, 5201280, 93260160, 1861574400,...}.

Examples

			{1},
{1, 1},
{1, 4, 1},
{1, 14, 14, 1},
{1, 44, 126, 44, 1},
{1, 132, 887, 887, 132, 1},
{1, 390, 5451, 12076, 5451, 390, 1},
{1, 1150, 30984, 131665, 131665, 30984, 1150, 1},
{1, 3400, 168076, 1252600, 2353126, 1252600, 168076, 3400, 1},
{1, 10088, 885725, 10905407, 34828859, 34828859, 10905407, 885725, 10088, 1},
{1, 30026, 4582497, 89401968, 454344414, 764856588, 454344414, 89401968, 4582497, 30026, 1}

Crossrefs

Cf. A008518, A060187

Programs

  • Mathematica
    p[x_, n_] = If[n == 0, 1, (1 - x)^(n + 1)*Sum[((2* k + 1)^n + (k + 1)^n + k^n)*x^k, {k, 0, Infinity}]/2];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]

Formula

p(x,n)=If[n == 0, 1, (1 - x)^(n + 1)*Sum[((2*k + 1)^n + (k + 1)^n + k^n)*x^k, {k, 0, Infinity}]/2];
t(n,m)=coefficients(p(x,n))=If[n==0,1,(A008518(n,m)+A060187(n,m))/2]
Previous Showing 11-16 of 16 results.

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

Content is available under The OEIS End-User License Agreement.