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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 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-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 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.)