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

0,4

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

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

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

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-François 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
(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

Crossrefs

Row sums A037952, matrix inverse A112552.

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

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