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.

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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

A320508 T(n,k) = binomial(n - k - 1, k), 0 <= k < n, and T(n,n) = (-1)^n, triangle read by rows.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 0, -1, 1, 2, 0, 0, 1, 1, 3, 1, 0, 0, -1, 1, 4, 3, 0, 0, 0, 1, 1, 5, 6, 1, 0, 0, 0, -1, 1, 6, 10, 4, 0, 0, 0, 0, 1, 1, 7, 15, 10, 1, 0, 0, 0, 0, -1, 1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, -1, 1, 10, 36
Offset: 0

Views

Author

Franck Maminirina Ramaharo, Oct 14 2018

Keywords

Comments

Differs from A164925 in signs.
The n-th row consists of the coefficients in the expansion of (-x)^n + (((1 + sqrt(1 + 4*x))/2)^n -((1 - sqrt(1 + 4*x))/2)^n )/sqrt(1 + 4*x).
The coefficients in the expansion of Sum_{j=0..floor((n - 1)/2)} T(n,k)*x^(n - 2*j - 1) yield the n-th row in A168561, the coefficients of the n-th Fibonacci polynomial.
Row n sums up to Fibonacci(n) + (-1)^n (A008346).

Examples

			Triangle begins:
    1;
    1, -1;
    1,  0,  1;
    1,  1,  0, -1;
    1,  2,  0,  0,  1;
    1,  3,  1,  0,  0, -1;
    1,  4,  3,  0,  0,  0, 1;
    1,  5,  6,  1,  0,  0, 0, -1;
    1,  6, 10,  4,  0,  0, 0,  0, 1;
    1,  7, 15, 10,  1,  0, 0,  0, 0, -1;
    1,  8, 21, 20,  5,  0, 0,  0, 0,  0, 1;
    1,  9, 28, 35, 15,  1, 0,  0, 0,  0, 0, -1;
    ...

Links

Crossrefs

Inspired by A123018.
Cf. A007318, A026729, A049310, A052553, A164925, A168561.

Programs

  • Mathematica
    Table[Table[Binomial[n - k - 1, k], {k, 0, n}], {n, 0, 12}]//Flatten
  • Maxima
    create_list(binomial(n - k - 1, k), n, 0, 12, k, 0, n);

Formula

G.f.: 1/((1 + x*y)*(1 - y - x*y^2)).
E.g.f.: exp(-x*y) + (exp(y*(1 + sqrt(1 + 4*x))/2) - exp(y*(1 - sqrt(1 + 4*x))/2))/sqrt(1 + 4*x).
T(n,1) = A023443(n).
Previous Showing 11-13 of 13 results.

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