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.

A176488 Triangle T(n,k) = A008292(n+1,k+1) + A176487(n,k) - 1, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 23, 23, 1, 1, 54, 136, 54, 1, 1, 117, 612, 612, 117, 1, 1, 244, 2395, 4850, 2395, 244, 1, 1, 499, 8605, 31271, 31271, 8605, 499, 1, 1, 1010, 29242, 176522, 312448, 176522, 29242, 1010, 1, 1, 2033, 95714, 910466, 2620832, 2620832, 910466
Offset: 0

Views

Author

Roger L. Bagula, Apr 19 2010

Keywords

Comments

Row sums are 1, 2, 10, 48, 246, 1460, 10130, 80752, 725998, 7258092, 79834602,....
Apparently the row sums obey (-45*n+124)*s(n) +(45*n^2+127*n-654)*s(n-1) +(-206*n^2+227*n+708)*s(n-2) +(303*n^2-869*n+458)*s(n-3) -2*(71*n-125)*(n-2)*s(n-4)=0. - R. J. Mathar, Jun 16 2015

Examples

			1;
1, 1;
1, 8, 1;
1, 23, 23, 1;
1, 54, 136, 54, 1;
1, 117, 612, 612, 117, 1;
1, 244, 2395, 4850, 2395, 244, 1;
1, 499, 8605, 31271, 31271, 8605, 499, 1;
1, 1010, 29242, 176522, 312448, 176522, 29242, 1010, 1;
1, 2033, 95714, 910466, 2620832, 2620832, 910466, 95714, 2033, 1;
1, 4080, 305317, 4407094, 19476436, 31448746, 19476436, 4407094, 305317, 4080, 1;

Crossrefs

Cf. A007318, A008292.

Programs

  • Maple
    A176488 := proc(n,k)
    A008292(n+1,k+1)+A176487(n,k)-1 ;
end proc: # R. J. Mathar, Jun 16 2015
  • Mathematica
    << DiscreteMath`Combinatorica`;
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Eulerian[1 + n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 1] + t[n, m, q - 2] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]

A176489 Triangle T(n,k) = A176487(n,k)+A176488(n,k)-1 read by rows 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 12, 1, 1, 35, 35, 1, 1, 82, 206, 82, 1, 1, 177, 922, 922, 177, 1, 1, 368, 3599, 7284, 3599, 368, 1, 1, 751, 12917, 46923, 46923, 12917, 751, 1, 1, 1518, 43876, 264810, 468706, 264810, 43876, 1518, 1, 1, 3053, 143588, 1365740, 3931310
Offset: 0

Views

Author

Roger L. Bagula, Apr 19 2010

Keywords

Comments

Row sums are 1, 2, 14, 72, 372, 2200, 15220, 121184, 1089116, 10887384, 119752404,....
The row sums s(n) seem to obey (-45*n+124)*s(n) +(45*n^2+127*n-654)*s(n-1) +(-206*n^2+227*n+708)*s(n-2) +(303*n^2-869*n+458)*s(n-3) -2*(71*n-125)*(n-2)*s(n-4)=0. - R. J. Mathar, Jun 16 2015

Examples

			1;
1, 1;
1, 12, 1;
1, 35, 35, 1;
1, 82, 206, 82, 1;
1, 177, 922, 922, 177, 1;
1, 368, 3599, 7284, 3599, 368, 1;
1, 751, 12917, 46923, 46923, 12917, 751, 1;
1, 1518, 43876, 264810, 468706, 264810, 43876, 1518, 1;
1, 3053, 143588, 1365740, 3931310, 3931310, 1365740, 143588, 3053, 1;
1, 6124, 457997, 6610700, 29214758, 47173244, 29214758, 6610700, 457997, 6124, 1;

Crossrefs

Cf. A007318, A008292.

Programs

  • Maple
    A176489 := proc(n,k)
    A176487(n,k)+A176488(n,k)-1 ;
end proc: # R. J. Mathar, Jun 16 2015
  • Mathematica
    << DiscreteMath`Combinatorica`;
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Eulerian[1 + n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 1] + t[n, m, q - 2] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]

A176490 Triangle T(n,k) = A008292(n+1,k+1) + A060187(n+1,k+1)- 1 read along rows 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 9, 1, 1, 33, 33, 1, 1, 101, 295, 101, 1, 1, 293, 1983, 1983, 293, 1, 1, 841, 11733, 25963, 11733, 841, 1, 1, 2425, 64949, 275341, 275341, 64949, 2425, 1, 1, 7053, 346219, 2573521, 4831203, 2573521, 346219, 7053, 1, 1, 20685, 1804179, 22163163
Offset: 0

Views

Author

Roger L. Bagula, Apr 19 2010

Keywords

Comments

Row sums are: 1, 2, 11, 68, 499, 4554, 51113, 685432, 10684791, 189423350, 3755807989,....
Conjecture on the row sums s(n): 859*(n+1)*s(n) +(-2577*n^2-15955*n+33324)*s(n-1) +(1718*n^3+39275*n^2-102106*n-16383)*s(n-2) +(-25038*n^3+35127*n^2+252701*n-453082)*s(n-3) +(n-3)*(57834*n^2-211893*n+212386)*s(n-4) -2*(17257*n-29530)*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 16 2015

Examples

			1;
1, 1;
1, 9, 1;
1, 33, 33, 1;
1, 101, 295, 101, 1;
1, 293, 1983, 1983, 293, 1;
1, 841, 11733, 25963, 11733, 841, 1;
1, 2425, 64949, 275341, 275341, 64949, 2425, 1;
1, 7053, 346219, 2573521, 4831203, 2573521, 346219, 7053, 1;
1, 20685, 1804179, 22163163, 70723647, 70723647, 22163163, 1804179, 20685, 1;
1, 61073, 9268777, 180504391, 916661395, 1542816715, 916661395, 180504391, 9268777, 61073, 1;

Crossrefs

Cf. A007318, A008292, A060187, A176487.

Programs

  • Maple
    A176490 := proc(n,k)
    A008292(n+1,k+1)+A060187(n+1,k+1)-1 ;
end proc: # R. J. Mathar, Jun 16 2015
  • Mathematica
    (*A060187*)
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
<< DiscreteMath`Combinatorica`;
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Eulerian[1 + n, m];
t[n_, m_, 2] := f[n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]

A176491 Triangle T(n,k) = binomial(n,k) + A176490(n,k) - 1 read along rows 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 35, 35, 1, 1, 104, 300, 104, 1, 1, 297, 1992, 1992, 297, 1, 1, 846, 11747, 25982, 11747, 846, 1, 1, 2431, 64969, 275375, 275375, 64969, 2431, 1, 1, 7060, 346246, 2573576, 4831272, 2573576, 346246, 7060, 1, 1, 20693, 1804214, 22163246
Offset: 0

Views

Author

Roger L. Bagula, Apr 19 2010

Keywords

Comments

Row sums are 1, 2, 12, 72, 510, 4580, 51170, 685552, 10685038, 189423852, 3755809002,....

Examples

			1;
1, 1;
1, 10, 1;
1, 35, 35, 1;
1, 104, 300, 104, 1;
1, 297, 1992, 1992, 297, 1;
1, 846, 11747, 25982, 11747, 846, 1;
1, 2431, 64969, 275375, 275375, 64969, 2431, 1;
1, 7060, 346246, 2573576, 4831272, 2573576, 346246, 7060, 1;
1, 20693, 1804214, 22163246, 70723772, 70723772, 22163246, 1804214, 20693, 1;
1, 61082, 9268821, 180504510, 916661604, 1542816966, 916661604, 180504510, 9268821, 61082, 1;

Crossrefs

Cf. A007318, A008292, A060187, A176487.

Programs

  • Maple
    A176491 := proc(n,k)
        A176490(n,k)+binomial(n,k)-1 ;
end proc: # R. J. Mathar, Jun 16 2015
  • Mathematica
    (*A060187*)
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
<< DiscreteMath`Combinatorica`;
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Eulerian[1 + n, m];
t[n_, m_, 2] := f[n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]

A176492 Triangle T(n,k) = A176492(n,k) + A008292(n+1,k+1) - 1 read along rows 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 13, 1, 1, 45, 45, 1, 1, 129, 365, 129, 1, 1, 353, 2293, 2293, 353, 1, 1, 965, 12937, 28397, 12937, 965, 1, 1, 2677, 69261, 290993, 290993, 69261, 2677, 1, 1, 7561, 360853, 2661809, 4987461, 2661809, 360853, 7561, 1, 1, 21705, 1852053, 22618437
Offset: 0

Views

Author

Roger L. Bagula, Apr 19 2010

Keywords

Comments

Row sums are 1, 2, 15, 92, 625, 5294, 56203, 725864, 11047909, 193052642, 3795725791,....

Examples

			1;
1, 1;
1, 13, 1;
1, 45, 45, 1;
1, 129, 365, 129, 1;
1, 353, 2293, 2293, 353, 1;
1, 965, 12937, 28397, 12937, 965, 1;
1, 2677, 69261, 290993, 290993, 69261, 2677, 1;
1, 7561, 360853, 2661809, 4987461, 2661809, 360853, 7561, 1;
, 21705, 1852053, 22618437, 72034125, 72034125, 22618437, 1852053, 21705, 1;
1, 63117, 9421457, 182707997, 926399717, 1558541213, 926399717, 182707997, 9421457, 63117, 1;

Crossrefs

Cf. A007318, A008292, A060187, A176487.

Programs

  • Maple
    A176492 := proc(n,k)
    A176491(n,k)+A008292(n+1,k+1)-1 ;
end proc: # R. J. Mathar, Jun 16 2015
  • Mathematica
    (*A060187*)
p[x_, n_] = (1 - x)^(n + 1)*Sum[(2*k + 1)^n*x^k, {k, 0, Infinity}];
f[n_, m_] := CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x][[m + 1]];
<< DiscreteMath`Combinatorica`;
t[n_, m_, 0] := Binomial[n, m];
t[n_, m_, 1] := Eulerian[1 + n, m];
t[n_, m_, 2] := f[n, m];
t[n_, m_, q_] := t[n, m, q] = t[n, m, q - 2] + t[n, m, q - 3] - 1;
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 0, 10}]
Showing 1-5 of 5 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.