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.

A114189 Riordan array (1/(1+xc(-2x)), xc(-2x)/(1+xc(-2x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, -1, 1, 3, -4, 1, -13, 19, -7, 1, 67, -102, 44, -10, 1, -381, 593, -278, 78, -13, 1, 2307, -3640, 1795, -568, 121, -16, 1, -14589, 23231, -11849, 4051, -999, 173, -19, 1, 95235, -152650, 79750, -28770, 7820, -1598, 234, -22, 1, -636925, 1025965, -545680, 204760, -59650, 13642, -2392, 304, -25, 1
Offset: 0

Views

Author

Paul Barry, Nov 16 2005

Keywords

Comments

Inverse of A114188. Factors as (1,xc(-2x))*(1/(1+x), x/(1+x)). Row sums are 0^n. Diagonal sums are A114190. First column is A114191. A signed version of A110506.
Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -1,-2,-2,-2,-2,-2,-2,...] DELTA [1,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 01 2007

Examples

			Triangle begins
     1;
    -1,    1;
     3,   -4,    1;
   -13,   19,   -7,   1;
    67, -102,   44, -10,   1;
  -381,  593, -278,  78, -13, 1;

Programs

  • Mathematica
    c[x_] := (1 - Sqrt[1 - 4x])/(2x);
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1 + # c[-2#])&, # c[-2#]/(1 + # c[-2#])&, 10] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Formula

Riordan array ((3-sqrt(1+8x))/(2(1-x)), (sqrt(1+8x)-2x-1)/(2(1-x))).
T(n,k) = (-1)^(n-k)*A110506(n,k). - Philippe Deléham, Mar 24 2007

A114193 Riordan array (1/(1+2xc(-2x)),xc(-2x)/(1+2xc(-2x))), c(x) the g.f. of A000108.

Original entry on oeis.org

1, -2, 1, 8, -6, 1, -40, 36, -10, 1, 224, -224, 80, -14, 1, -1344, 1440, -600, 140, -18, 1, 8448, -9504, 4400, -1232, 216, -22, 1, -54912, 64064, -32032, 10192, -2184, 308, -26, 1, 366080, -439296, 232960, -81536, 20160, -3520, 416, -30, 1, -2489344, 3055104, -1697280, 639744, -176256, 35904, -5304, 540, -34, 1
Offset: 0

Views

Author

Paul Barry, Nov 16 2005

Keywords

Comments

Row sums are A114191. Diagonal sums are A114194. Inverse of A114192.
Triangle T(n,k), read by rows, given by (-2, -2, -2, -2, -2, -2, -2, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 26 2014

Examples

			Triangle begins
      1;
     -2,    1;
      8,   -6,    1;
    -40,   36,  -10,   1;
    224, -224,   80, -14,   1;
  -1344, 1440, -600, 140, -18, 1;

Crossrefs

Cf. A039599, A084938.

Programs

  • Mathematica
    c[x_] := (1 - Sqrt[1 - 4 x])/(2 x);
(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1 + 2 # c[-2 #])&, # c[-2 #]/(1 + 2 # c[-2 #])&, 10] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Formula

Riordan array ((sqrt(1+8x)-1)/(4x), (sqrt(1+8x)-1)^2/(16x)).
T(n, k) = (-2)^(n-k)*A039599(n, k) = (-2)^(n-k)*C(2*n, n-k)*(2*k+1)/(n+k+1). - Philippe Deléham, Nov 17 2005

A166078 Expansion of (3(1-x)-sqrt(1+6x-7x^2))/(2(1-x)(1-2x)).

Original entry on oeis.org

1, 0, 2, -6, 30, -150, 806, -4494, 25822, -151782, 908502, -5518590, 33933774, -210814422, 1321230150, -8343458286, 53037407166, -339111023046, 2179407749558, -14071216784862, 91225811704750, -593639364476598
Offset: 0

Views

Author

Paul Barry, Oct 06 2009

Keywords

Comments

First column of inverse of Riordan array ((1+x)/(1+x+2x^2),x(1+2x)/(1+x+2x^2)).
Hankel transform is A002416. Binomial transform of A114191.

References

  • A. Hora, N. Obata, Quantum Probability and Spectral Analysis of Graphs, Springer, 2007, p. 122

Links

Programs

  • Mathematica
    CoefficientList[Series[(3 (1 - x) - Sqrt[1 + 6 x - 7 x^2])/(2 (1 - x) (1 - 2 x)), {x, 0, 50}], x](* G. C. Greubel, Apr 24 2016 *)

Formula

G.f.: 1/(1-x+x*c(-2x/(1-x))), c(x) the g.f. of A000108.
G.f.: 1/(1-2x^2/(1-3x-4x^2/(1-3x-4x^2/(1-3x-4x^2/(1-... (continued fraction).
Conjecture: n*a(n) + 2*(2*n-5)*a(n-1) + (34-19*n)*a(n-2) + 14*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 14 2011
Showing 1-3 of 3 results.

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