A112554 Riordan array (c(x^2)^2, x*c(x^2)), c(x) the g.f. of A000108.

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

Offset: 0

Author

Paul Barry, Sep 13 2005

Keywords

• easy
• nonn
• tabl

Comments

Inverse of A112552.

The n-th row polynomial (in descending powers of x) is equal to the n-th degree Taylor polynomial of the polynomial 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

Examples

			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;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Peter Bala, A 4-parameter family of embedded Riordan arrays

Crossrefs

Row sums A037952, matrix inverse A112552.

Cf. A000108, A037952 (row sums), A112552, A112553.

Programs

Magma

A112554:= func< n,k | ((1+(-1)^(n-k))/2)*(Binomial(n, Floor((n-k)/2)) - Binomial(n, Floor((n-k-4)/2))) >;
[A112554(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 13 2022

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-François Alcover, Jun 13 2019 *)

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

Formula

Sum_{k=0..n} T(n, k) = binomial(n+1, floor(n/2)) = A037952(n+1).

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