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.

A109247 Expansion of (1 - 3*x^2 - 3*x^3 + x^4)/(1 + x^4).

Original entry on oeis.org

1, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3, 0, 0, 3, 3, 0, 0, -3, -3
Offset: 0

Views

Author

Paul Barry, Jun 23 2005

Keywords

Comments

Row sums of Riordan array (1-x-2x^2,x(1-x)), A109246.
After the initial terms, cyclic with period 8: [0,0,-3,-3,0,0,3,3]. - Antti Karttunen, Aug 12 2017

Links

Crossrefs

Cf. A109246, A132380.

Programs

  • Mathematica
    CoefficientList[Series[(1-3x^2-3x^3+x^4)/(1+x^4),{x,0,90}],x] (* or *) Join[{1},LinearRecurrence[{0,0,0,-1},{0,-3,-3,0},90]] (* Harvey P. Dale, Mar 24 2012 *)
  • PARI
    Vec((1-3*x^2-3*x^3+x^4)/(1+x^4) + O(x^80)) \\ Jinyuan Wang, Mar 22 2020
  • Scheme
    (define (A109247 n) (case n ((0) 1) ((1 4) 0) ((2 3) -3) (else (- (A109247 (- n 4)))))) ;; (After Harvey P. Dale's Mar 24 2012 recurrence) - Antti Karttunen, Aug 12 2017

Formula

a(0)=1, a(1)=0, a(2)=-3, a(3)=-3, a(4)=0, a(n)=-a(n-4) - Harvey P. Dale, Mar 24 2012
For n > 0, a(n) = -3 * A132380(n). - Antti Karttunen, Aug 12 2017

A109244 A tree-node counting triangle.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 13, 7, 3, 1, 46, 24, 11, 4, 1, 166, 86, 40, 16, 5, 1, 610, 314, 148, 62, 22, 6, 1, 2269, 1163, 553, 239, 91, 29, 7, 1, 8518, 4352, 2083, 920, 367, 128, 37, 8, 1, 32206, 16414, 7896, 3544, 1461, 541, 174, 46, 9, 1, 122464, 62292, 30086, 13672, 5776, 2232
Offset: 0

Views

Author

Paul Barry, Jun 23 2005

Keywords

Comments

Columns include A026641,A014300,A014301. Inverse matrix is A109246. Row sums are A014300. Diagonal sums are A109245.

Examples

			Rows begin:
  1;
  1,1;
  4,2,1;
  13,7,3,1;
  46,24,11,4,1;
  166,86,40,16,5,1;

Links

Programs

  • GAP
    Flat(List([0..12], n-> List([0..n], k-> Sum([0..n], j-> (-1)^(n-j)*Binomial(n+j-k, j-k) )))); # G. C. Greubel, Feb 19 2019
  • Magma
    [[(&+[(-1)^(n-j)*Binomial(n+j-k, j-k): j in [0..n]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Feb 19 2019
  • Mathematica
    Table[Sum[(-1)^(n-j)*Binomial[n+j-k, j-k], {j,0,n}], {n,0,12}, {k,0,n}] //Flatten  (* G. C. Greubel, Feb 19 2019 *)
  • PARI
    {T(n,k) = sum(j=0,n, (-1)^(n-j)*binomial(n+j-k, j-k))};
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 19 2019
  • Sage
    [[sum((-1)^(n-j)*binomial(n+j-k, j-k) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Feb 19 2019

Formula

Number triangle T(n, k) = Sum_{i=0..n} (-1)^(n-i)*binomial(n+i-k, i-k).
Riordan array (1/(1-x*c(x)-2*x^2*c(x)^2), x*c(x)) where c(x)=g.f. of A000108.
The production matrix M (discarding the zeros) is:
1, 1;
3, 1, 1;
3, 1, 1, 1;
3, 1, 1, 1, 1;
... such that the n-th row of the triangle is the top row of M^n. - Gary W. Adamson, Feb 16 2012

A109248 Expansion of (1-x-2*x^2)/(1-x^2+x^3).

Original entry on oeis.org

1, -1, -1, -2, 0, -1, 2, -1, 3, -3, 4, -6, 7, -10, 13, -17, 23, -30, 40, -53, 70, -93, 123, -163, 216, -286, 379, -502, 665, -881, 1167, -1546, 2048, -2713, 3594, -4761, 6307, -8355, 11068, -14662, 19423, -25730, 34085, -45153, 59815, -79238, 104968, -139053, 184206, -244021, 323259, -428227, 567280
Offset: 0

Views

Author

Paul Barry, Jun 23 2005

Keywords

Comments

Diagonal sums of Riordan array (1-x-2x^2,x(1-x)), A109246.

Links

Programs

  • Magma
    I:=[1,-1,-1,-2,0,-1]; [n le 6 select I[n] else Self(n-2)-Self(n-3): n in [1..60]]; // Vincenzo Librandi, Mar 12 2014
  • Mathematica
    a[0] = 1; a[1] = -1; a[2] = -1; a[n_] := a[n - 2] - a[n - 3]; Table[a[n], {n, 0, 50}] (* Wesley Ivan Hurt, Mar 06 2014 *)
CoefficientList[Series[(1 - x - 2 x^2)/(1 - x^2 + x^3), {x, 0, 40}], x] (* Vincenzo Librandi. Mar 12 2014 *)
LinearRecurrence[{0,1,-1},{1,-1,-1},60] (* Harvey P. Dale, Jun 03 2014 *)
  • PARI
    Vec((1-x-2*x^2)/(1-x^2+x^3) + O(x^50)) \\ Michel Marcus, Sep 17 2016

Formula

a(n) = a(n-2) - a(n-3), starting 1, -1, -1.
a(n) = (-1)^n * (A000931(n) - A000931(n-3) ), for n>2. - Ralf Stephan, Mar 10 2014
Showing 1-3 of 3 results.

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