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A112554 Riordan array (c(x^2)^2, xc(x^2)), c(x) the g.f. of A000108. 3
1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 5, 0, 4, 0, 1, 0, 9, 0, 5, 0, 1, 14, 0, 14, 0, 6, 0, 1, 0, 28, 0, 20, 0, 7, 0, 1, 42, 0, 48, 0, 27, 0, 8, 0, 1, 0, 90, 0, 75, 0, 35, 0, 9, 0, 1, 132, 0, 165, 0, 110, 0, 44, 0, 10, 0, 1, 0, 297, 0, 275, 0, 154, 0, 54, 0, 11, 0, 1, 429, 0, 572, 0, 429, 0, 208, 0, 65, 0, 12, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Inverse of A112552. Row sums are C(n+1,floor(n/2)), A037952(n+1).

The n-th row polynomial (in descending powers of x) is equal to the n-th degree Taylor polynomial of the rational function (1 - x^4)*(1 + x^2)^n about 0. For example, when n = 6,  (1 - x^4)*(1 + x^2)^6 = 1 + 6*x^2 + 14*x^4 + 14*x^6 + O(x^8). - Peter Bala, Feb 19 2018

LINKS

Table of n, a(n) for n=0..90.

Peter Bala, A 4-parameter family of embedded Riordan arrays

FORMULA

T(n,k) = (1 + (-1)^(n-k))/2*binomial(n, floor((n - k)/2)) - binomial(n, floor((n - k - 4)/2 )). - Peter Bala, Feb 19 2018

EXAMPLE

Triangle begins

   1;

   0, 1;

   2, 0,  1;

   0, 3,  0, 1;

   5, 0,  4, 0, 1;

   0, 9,  0, 5, 0, 1;

  14, 0, 14, 0, 6, 0, 1;

MAPLE

seq(seq((1 + (-1)^(n-k))/2*( binomial(n, floor((n - k)/2)) - binomial(n, floor((n - k - 4)/2 )) ), k = 0..n), n = 0..10); # Peter Bala, Feb 19 2018

MATHEMATICA

T[n_, k_] := (1 + (-1)^(n-k))/2 (Binomial[n, Floor[(n-k)/2]] - Binomial[n, Floor[(n-k-4)/2]]);

Table[T[n, k], {n, 0, 12}, {k, 0, n}] (* Jean-Fran├žois Alcover, Jun 13 2019 *)

PROG

(Sage) # Algorithm of L. Seidel (1877)

# Prints the first n rows of a signed version of the triangle.

def Signed_A112554_triangle(n) :

    D = [0]*(n+4); D[1] = 1

    b = False; h = 2

    for i in range(2*n+2) :

        if b :

            for k in range(h, 0, -1) : D[k] += D[k-1]

            h += 1

        else :

            for k in range(1, h, 1) : D[k] -= D[k+1]

        b = not b

        if b and i > 0 : print([D[z] for z in (2..h-1)])

Signed_A112554_triangle(13) # Peter Luschny, May 01 2012

CROSSREFS

Row sums A037952, matrix inverse A112552. Cf. A000108.

Sequence in context: A048154 A320602 A134511 * A120616 A108044 A104477

Adjacent sequences:  A112551 A112552 A112553 * A112555 A112556 A112557

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Sep 13 2005

STATUS

approved

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Last modified April 21 15:55 EDT 2021. Contains 343156 sequences. (Running on oeis4.)