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 1-3 of 3 results.

A141720 Triangle of coefficients of the polynomials (1 - x)^n*A(n,x/(1 - x)), where A(n,x) are the Eulerian polynomials of A008292.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, -2, 0, 1, 8, -8, 0, 1, 22, -6, -32, 16, 0, 1, 52, 84, -272, 136, 0, 1, 114, 606, -1168, -96, 816, -272, 0, 1, 240, 2832, -2176, -8832, 11904, -3968, 0, 1, 494, 11122, 11072, -83360, 71168, 13312, -31744, 7936, 0, 1, 1004, 39772, 148592, -472760, -17152, 831232, -707584, 176896
Offset: 1

Views

Author

Roger L. Bagula, Sep 11 2008

Keywords

Comments

Row sums are one.
Row n gives the coefficients in the expansion of Sum_{j=1..n} A008292(n,j)*x^j*(1 - x)^(n - j).
The coefficients of the polynomials (1 + x)^n*A(n,x/(1 + x)) are listed in A019538.

Examples

			Triangle begins:
  0, 1;
  0, 1;
  0, 1,    2,    -2;
  0, 1,    8,    -8;
  0, 1,   22,    -6,    -32,      16;
  0, 1,   52,    84,   -272,     136;
  0, 1,  114,   606,  -1168,     -96,    816,   -272;
  0, 1,  240,  2832,  -2176,   -8832,  11904,  -3968;
  0, 1,  494, 11122,  11072,  -83360,  71168,  13312,  -31744,   7936;
  0, 1, 1004, 39772, 148592, -472760, -17152, 831232, -707584, 176896;
  ...

Links

Crossrefs

Cf. A008292, A019538, A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A144387, A144400, A174128.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
f:= func< n,x | n eq 0 select 1 else (&+[EulerianNumber(n,j-1)*x^j*(1-x)^(n-j): j in [1..n]]) >;
A141720:= func< n,k | Coefficient(R!( f(n,x) ), k) >;
[A141720(n,k): k in [0..2*Floor((n+1)/2)-1], n in [1..15]]; // G. C. Greubel, Dec 30 2024
  • Maple
    CL := p -> PolynomialTools:-CoefficientList(p,x): flatten := seq -> ListTools:-Flatten(seq): flatten([seq(CL(add(A008292(n,j)*x^j*(1-x)^(n-j), j=1..n)), n=1..10)]); # Peter Luschny, Oct 25 2018
  • Mathematica
    Table[CoefficientList[FullSimplify[(1-2x)^(1+n)*PolyLog[-n, x/(1-x)]/(1-x)], x], {n, 1, 10}]//Flatten
  • Sage
    def A(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j)^n for j in (0..k))
def p(n,x): return sum( A(n, j)*x^j*(1-x)^(n-j) for j in (0..n) )
def A141720(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
flatten([A141720(n) for n in range(1,13)]) # G. C. Greubel, Jul 15 2021

Formula

Row n is generated by the polynomial (1 - 2*x)^(n+1)*Li(-n, x/(1-x))/(1 - x), where Li(n, z) is the polylogarithm function.
Also generated by Sum_{k=0..n} (eulerian(n,k)*Sum_{l=0..n} (-1)^l*(n - l + 1)*(2 - x)^l*C(l + 1, k)). - Mourad Rahmani (mrahmani(AT)usthb.dz), Jul 22 2010
E.g.f.: (x*exp(2*x*y) - x*exp(y))/(x*exp(y) - (1 - x)*exp(2*x*y)). - Franck Maminirina Ramaharo, Oct 24 2018

Extensions

Edited by Peter Bala, Jul 04 2012
Edited, and extra term removed by Franck Maminirina Ramaharo, Oct 24 2018

A144400 Triangle read by rows: row n (n > 0) gives the coefficients of x^k (0 <= k <= n - 1) in the expansion of Sum_{j=0..n} A000931(j+4)*binomial(n, j)*x^(j - 1)*(1 - x)^(n - j).

Original entry on oeis.org

1, 2, -1, 3, -3, 1, 4, -6, 4, 0, 5, -10, 10, 0, -3, 6, -15, 20, 0, -18, 10, 7, -21, 35, 0, -63, 70, -24, 8, -28, 56, 0, -168, 280, -192, 49, 9, -36, 84, 0, -378, 840, -864, 441, -89, 10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145, 11, -55, 165, 0
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 03 2008

Keywords

Examples

			Triangle begins:
    1;
    2,  -1;
    3,  -3,   1;
    4,  -6,   4, 0;
    5, -10,  10, 0,   -3;
    6, -15,  20, 0,  -18,   10;
    7, -21,  35, 0,  -63,   70,   -24;
    8, -28,  56, 0, -168,  280,  -192,   49;
    9, -36,  84, 0, -378,  840,  -864,  441,  -89;
   10, -45, 120, 0, -756, 2100, -2880, 2205, -890, 145;
     ... reformatted. - _Franck Maminirina Ramaharo_, Oct 22 2018

Links

Crossrefs

Row sums: subsequence of A000931, A078027, A182097.
Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A174128.

Programs

  • Mathematica
    a[n_]:= a[n]= If[n<3, Fibonacci[n], a[n-2] + a[n-3]];
p[x_, n_]:= Sum[a[k]*Binomial[n, k]*x^(k-1)*(1-x)^(n-k), {k, 0, n}];
Table[Coefficient[p[x, n], x, k], {n, 12}, {k, 0, n-1}]//Flatten
  • Sage
    @CachedFunction
def f(n): return fibonacci(n) if (n<3) else f(n-2) + f(n-3)
def p(n,x): return sum( binomial(n,j)*f(j)*x^(j-1)*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (1..12)] # G. C. Greubel, Jul 14 2021

Formula

G.f.: (y - (1 - 2*x)*y^2)/(1 - 3*(1 - x)*y + (3 - 6*x + 2*x^2)*y^2 - (1 - 3*x + 2*x^2 + x^3)*y^3). - Franck Maminirina Ramaharo, Oct 22 2018

Extensions

Edited, and new name by Franck Maminirina Ramaharo, Oct 22 2018

A174128 Irregular triangle read by rows: row n (n > 0) is the expansion of Sum_{m=1..n} A001263(n,m)*x^(m - 1)*(1 - x)^(n - m).

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 3, -3, 1, 6, -4, -4, 2, 1, 10, 0, -20, 10, 1, 15, 15, -55, 15, 15, -5, 1, 21, 49, -105, -35, 105, -35, 1, 28, 112, -140, -266, 364, -56, -56, 14, 1, 36, 216, -84, -882, 756, 336, -504, 126, 1, 45, 375, 210, -2100, 672, 2520, -2100, 210, 210, -42
Offset: 1

Views

Author

Roger L. Bagula, Mar 09 2010

Keywords

Comments

Row n gives the coefficients in the expansion of (1/x)*(1 - x)^n*N(n,x/(1 - x)), where N(n,x) is the n-th row polynomial for the triangle of Narayana numbers A001263.

Examples

			Triangle begins
    1;
    1;
    1,  1,  -1;
    1,  3,  -3;
    1,  6,  -4,   -4,    2;
    1, 10,   0,  -20,   10;
    1, 15,  15,  -55,   15,  15,  -5;
    1, 21,  49, -105,  -35, 105, -35;
    1, 28, 112, -140, -266, 364, -56,  -56,  14;
    1, 36, 216,  -84, -882, 756, 336, -504, 126;
    ...

Links

Crossrefs

Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144387, A144400.

Programs

  • Mathematica
    p[x_, n_]:= p[x, n]= Sum[(Binomial[n, j]*Binomial[n, j-1]/n)*x^j*(1-x)^(n-j), {j, 1, n}]/x;
Table[CoefficientList[p[x, n], x], {n, 1, 12}]//Flatten
  • Sage
    def p(n,x): return (1/(n*x))*sum( binomial(n,j)*binomial(n,j-1)*x^j*(1-x)^(n-j) for j in (1..n) )
def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (1..12)] # G. C. Greubel, Jul 14 2021

Formula

The n-th row of the triangle is generated by the coefficients of (1 - x)^(n - 1)*F(-n, 1 - n; 2; x/(1 - x)), where F(a, b ; c; z) is the ordinary hypergeometric function.
G.f.: (1 - y - sqrt(1 - 2*y + ((1 - 2*x)*y)^2))/(2*(1 - x)*x*y). - Franck Maminirina Ramaharo, Oct 23 2018

Extensions

Edited and new name by Joerg Arndt, Oct 28 2014
Comments and formula clarified by Franck Maminirina Ramaharo, Oct 23 2018
Showing 1-3 of 3 results.

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