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A099363 An inverse Chebyshev transform of 1-x. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

LINKS

Table of n, a(n) for n=0..38.

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

PROG

(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

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

CROSSREFS

Cf. A000245.

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

Sequence in context: A129996 A208355 * A106181 A098887 A259097 A245845

Adjacent sequences:  A099360 A099361 A099362 * A099364 A099365 A099366

KEYWORD

easy,sign

AUTHOR

Paul Barry, Oct 13 2004

STATUS

approved

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Last modified October 15 19:24 EDT 2018. Contains 316237 sequences. (Running on oeis4.)