cp's OEIS Frontend

A099363 An inverse Chebyshev transform of 1-x.

%I A099363 #23 Jun 11 2019 07:00:29
%S A099363 1,-1,1,-2,2,-5,5,-14,14,-42,42,-132,132,-429,429,-1430,1430,-4862,
%T A099363 4862,-16796,16796,-58786,58786,-208012,208012,-742900,742900,
%U A099363 -2674440,2674440,-9694845,9694845,-35357670,35357670,-129644790,129644790,-477638700,477638700,-1767263190,1767263190
%N A099363 An inverse Chebyshev transform of 1-x.
%C A099363 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)).
%C A099363 A208355(n) = abs(a(n)). - _Reinhard Zumkeller_, Mar 03 2012
%F A099363 G.f.: (1-(1-x)c(x^2))/x where c(x) is the g.f. of the Catalan numbers A000108.
%F A099363 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)}.
%F A099363 a(n) = (-1)^n A208355(n) = (-1)^n A000108([(n+1)/2]): Repeated Catalan numbers with alternating sign. - _M. F. Hasler_, Aug 25 2012
%F A099363 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
%t A099363 Table[(-1)^n CatalanNumber[Floor[(n+1)/2]], {n, 0, 38}] (* _Jean-François Alcover_, Jun 11 2019 *)
%o A099363 (Sage)
%o A099363 def A099363_list(n) :
%o A099363     D = [0]*(n+2); D[1] = 1
%o A099363     b = True; h = 2; R = []
%o A099363     for i in range(2*n-1) :
%o A099363         if b :
%o A099363             for k in range(h,0,-1) : D[k] -= D[k-1]
%o A099363             h += 1; R.append((-1)^(h//2)*D[2])
%o A099363         else :
%o A099363             for k in range(1,h, 1) : D[k] += D[k+1]
%o A099363         b = not b
%o A099363     return R
%o A099363 A099363_list(39) # _Peter Luschny_, Jun 03 2012
%o A099363 (PARI) A099363(n)=(-1)^n*A000108((n+1)\2) \\ _M. F. Hasler_, Aug 25 2012
%Y A099363 Cf. A000245.
%Y A099363 Cf. A106181, A129996 and A208355, which also consist of duplicated Catalan numbers A000108. - _M. F. Hasler_, Aug 25 2012
%K A099363 easy,sign
%O A099363 0,4
%A A099363 _Paul Barry_, Oct 13 2004