A099363 An inverse Chebyshev transform of 1-x.

1, -1, 1, -2, 2, -5, 5, -14, 14, -42, 42, -132, 132, -429, 429, -1430, 1430, -4862, 4862, -16796, 16796, -58786, 58786, -208012, 208012, -742900, 742900, -2674440, 2674440, -9694845, 9694845, -35357670, 35357670, -129644790, 129644790, -477638700, 477638700, -1767263190, 1767263190

Offset: 0

Author

Paul Barry, Oct 13 2004

Keywords

• easy
• sign

Comments

Second binomial transform of the expansion of c(-x)^3. The g.f. is transformed to 1-x under the Chebyshev transformation A(x)->(1/(1+x^2))A(x/(1+x^2)).

A208355(n) = abs(a(n)). - Reinhard Zumkeller, Mar 03 2012

Crossrefs

Cf. A000245.

Cf. A106181, A129996 and A208355, which also consist of duplicated Catalan numbers A000108. - M. F. Hasler, Aug 25 2012

Programs

Mathematica

Table[(-1)^n CatalanNumber[Floor[(n+1)/2]], {n, 0, 38}] (* Jean-François Alcover, Jun 11 2019 *)

PARI

A099363(n)=(-1)^n*A000108((n+1)\2) \\ M. F. Hasler, Aug 25 2012

Sage

def A099363_list(n) :
    D = [0]*(n+2); D[1] = 1
    b = True; h = 2; R = []
    for i in range(2*n-1) :
        if b :
            for k in range(h,0,-1) : D[k] -= D[k-1]
            h += 1; R.append((-1)^(h//2)*D[2])
        else :
            for k in range(1,h, 1) : D[k] += D[k+1]
        b = not b
    return R
A099363_list(39) # Peter Luschny, Jun 03 2012

Formula

G.f.: (1-(1-x)c(x^2))/x where c(x) is the g.f. of the Catalan numbers A000108.

a(n) = sum{k=0..n, (k+1)C(n, (n-k)/2)(0^k-sum{j=0..k, C(k, j)(-1)^(k-j)*j})(1+(-1)^(n-k))/(n+k+2)}.

a(n) = (-1)^n A208355(n) = (-1)^n A000108([(n+1)/2]): Repeated Catalan numbers with alternating sign. - M. F. Hasler, Aug 25 2012

Conjecture: (n+3)*a(n) +(-n-1)*a(n-1) -4*n*a(n-2) +4*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 26 2012