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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A116949 Riordan array ((1-x^3)/(1+2x^2),x).

Original entry on oeis.org

1, 0, 1, -2, 0, 1, -1, -2, 0, 1, 4, -1, -2, 0, 1, 2, 4, -1, -2, 0, 1, -8, 2, 4, -1, -2, 0, 1, -4, -8, 2, 4, -1, -2, 0, 1, 16, -4, -8, 2, 4, -1, -2, 0, 1, 8, 16, -4, -8, 2, 4, -1, -2, 0, 1, -32, 8, 16, -4, -8, 2, 4, -1, -2, 0, 1, -16, -32, 8, 16, -4, -8, 2, 4, -1, -2, 0, 1, 64, -16, -32, 8, 16, -4, -8, 2, 4, -1, -2, 0, 1
Offset: 0

Views

Author

Paul Barry, Mar 29 2006

Keywords

Comments

Sequence array of (-2)^(n/2)(1+(-1)^n)/2-(-2)^((n-3)/2)(1-(-1)^n)/2-(comb(1,n)-comb(0,n))/2.
Inverse of A116948.
Row sums are A117575. Diagonal sums are A117576.

Examples

			Number triangle begins
1,
0, 1,
-2, 0, 1,
-1, -2, 0, 1,
4, -1, -2, 0, 1,
2, 4, -1, -2, 0, 1,
-8, 2, 4, -1, -2, 0, 1,
-4, -8, 2, 4, -1, -2, 0, 1,
16, -4, -8, 2, 4, -1, -2, 0, 1

Crossrefs

Cf. A117575, A117576, A116948.

Extensions

Entry revised by N. J. A. Sloane, Apr 10 2006

A122016 Riordan array(1, x*(1+2*x)/(1-x)).

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 3, 6, 1, 0, 3, 15, 9, 1, 0, 3, 24, 36, 12, 1, 0, 3, 33, 90, 66, 15, 1, 0, 3, 42, 171, 228, 105, 18, 1, 0, 3, 51, 279, 579, 465, 153, 21, 1, 0, 3, 60, 414, 1200, 1500, 828, 210, 24, 1, 0, 3, 69, 576, 2172, 3858, 3258, 1344, 276, 27, 1
Offset: 0

Views

Author

Philippe Deléham, Sep 24 2006

Keywords

Comments

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,3,-2,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. Rising and falling diagonals are A078010 and A122552.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 3,  1;
  0, 3,  6,   1;
  0, 3, 15,   9,    1;
  0, 3, 24,  36,   12,    1;
  0, 3, 33,  90,   66,   15,   1;
  0, 3, 42, 171,  228,  105,  18,   1;
  0, 3, 51, 279,  579,  465, 153,  21,  1;
  0, 3, 60, 414, 1200, 1500, 828, 210, 24, 1;

Links

Crossrefs

Cf. Diagonals A000012, A008585, A062741, columns A000007, A122553, A122709, row sums A026150.
Cf. A078010, A084938, A102900, A117575, A122117, A122552, A147518.

Programs

  • Mathematica
    T[n_,k_]:=SeriesCoefficient[(1-x)/(1-(y+1)*x-2*y*x^2),{x,0,n},{y,0,k}]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Dec 27 2023 *)

Formula

Sum_{k=0..n} T(n,k)*x^(n-k) = A026150(n), A102900(n) for x = 1, 2.
T(n,k) = T(n-1,k) + T(n-1,k-1) + 2*T(n-2,k-1). - Philippe Deléham, Sep 25 2006
G.f.: (1-x)/(1-(y+1)*x-2*y*x^2). - Philippe Deléham, Jan 31 2012
Sum_{k=0..n} T(n,k)*x^k = A117575(n+1), A000007(n), A026150(n), A122117(n), A147518(n) for x = -1, 0, 1, 2, 3 respectively. - Philippe Deléham, Jan 31 2012

Extensions

More terms from Stefano Spezia, Dec 27 2023

A280193 a(2*n) = 2, a(2*n + 1) = -1, a(0) = 1.

Original entry on oeis.org

1, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2, -1, 2
Offset: 0

Views

Author

Michael Somos, Dec 28 2016

Keywords

Examples

			G.f. = 1 - x + 2*x^2 - x^3 + 2*x^4 - x^5 + 2*x^6 - x^7 + 2*x^8 - x^9 + ...

Links

Crossrefs

Cf. A000225, A000325, A040001, A028242, A079583, A083329, A117575.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 - x+x^2)/(1-x^2))); // G. C. Greubel, Jul 29 2018
  • Mathematica
    a[ n_] := Which[ n < 1, Boole[n == 0], OddQ[n], -1, True, 2];
a[ n_] := SeriesCoefficient[ (1 - x + x^2) / (1 - x^2), {x, 0, n}];
LinearRecurrence[{0,1},{1,-1,2},80] (* Harvey P. Dale, Aug 06 2025 *)
  • PARI
    {a(n) = if( n<1, n==0, 2 - 3*(n%2))};
  • PARI
    {a(n) = if( n<1, n==0, [2, -1][n%2 + 1])};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( (1 - x + x^2) / (1 - x^2) + x * O(x^n), n))};

Formula

Euler transform of length 6 sequence [-1, 2, 1, 0, 0, -1].
Moebius transform is length 2 sequence [-1, 3].
a(n) = -b(n) where b() is multiplicative with b(2^e) = -2 if e>0, b(p^e) = 1 otherwise.
G.f.: (1 - x + x^2) / (1 - x^2).
G.f.: (1 - x) * (1 - x^6) / ((1 - x^3) * (1 -x^2)^2).
G.f.: 1 / (1 + x / (1 + x / (1 - 3*x / (1 + x)))).
a(n) = (-1)^n * A040001(n).
A028242(n) = Sum_{k=0..n} a(k).
A117575(n+1) = Product_{k=0..n} a(k).
A000225(n-1) = Sum_{k=0..n} binomial(n, k) * a(k) if n>0.
A000325(n) = Sum_{k=0..n} binomial(n, k+1) * a(k) if n>0.
a(n) = Sum_{k=0..n} binomial(n, k) * (-1)^k * A083329(k).
A079583(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n.
a(n) = A168361(n+1), n>0. - R. J. Mathar, Jan 04 2017
Previous Showing 21-23 of 23 results.

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

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