A114188 -id:A114188 - cp's OEIS frontend

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

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

Paul Barry, Nov 16 2005

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

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

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 *)

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

A114192 Riordan array (1/(1-2x),x/(1-2x)^2).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 24, 10, 1, 16, 80, 60, 14, 1, 32, 240, 280, 112, 18, 1, 64, 672, 1120, 672, 180, 22, 1, 128, 1792, 4032, 3360, 1320, 264, 26, 1, 256, 4608, 13440, 14784, 7920, 2288, 364, 30, 1, 512, 11520, 42240, 59136, 41184, 16016, 3640, 480, 34, 1

Offset: 0

Paul Barry, Nov 16 2005

Factors as (1/(1-x),x/(1-x))*(1/(1-x),x*(1+x)/(1-x)^2) or A007318 times A114188. Also (1/(1-2*x),x/(1-2*x))*(1,x*(1+2*x)). Inverse is A114193. Row sums are A007583. Diagonal sums are A007051.

			Triangle begins
1;
2, 1;
4, 6, 1;
8, 24, 10, 1;
16, 80, 60, 14, 1;
32, 240, 280, 112, 18, 1;

T(n,k) = sum{j=0..n, C(k, j)*C(n, k+j)}*2^(n-k).
T(n,k) = 2^(n-k)*binomial(n+k,2k) = 2^(n-k)*A085478(n,k). - Philippe Deléham, May 05 2006
T(n,k) = A013609(n+k, n-k). - Johannes W. Meijer, Sep 05 2013
T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - 4*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = 2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 17 2014

cp's OEIS Frontend

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

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