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A064306 Convolution of A052701 (Catalan numbers multiplied by powers of 2) with powers of -1.

%I A064306 #22 Jan 05 2025 19:51:36
%S A064306 1,1,7,33,191,1153,7295,47617,318463,2170881,15028223,105365505,
%T A064306 746651647,5339185153,38478839807,279201841153,2037998419967,
%U A064306 14954803494913,110255315877887,816299567480833,6066679566041087
%N A064306 Convolution of A052701 (Catalan numbers multiplied by powers of 2) with powers of -1.
%H A064306 G. C. Greubel, <a href="/A064306/b064306.txt">Table of n, a(n) for n = 0..1000</a>
%H A064306 W. Lang, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/40-4/lang.pdf">On polynomials related to derivatives of the generating function of Catalan numbers</a>, Fib. Quart. 40,4 (2002) 299-313; Eq.(31) with lambda=-1/2.
%F A064306 a(n) = (-1)^n*Sum_{k=0,..,n} (C(k)/(-1/2)^k) with C(k)=A000108(k) (Catalan).
%F A064306 a(n) = -a(n-1) + C(n)*2^n, n >= 0, a(-1) := 0, with C(n)=A000108(n).
%F A064306 G.f.: A(2*x)/(1+x), with A(x) g.f. of Catalan numbers A000108.
%F A064306 Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Dec 09 2013
%F A064306 a(n) ~ 2^(3*n+3)/(9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Dec 09 2013
%t A064306 CoefficientList[Series[(1-Sqrt[1-8*x])/(4*x*(1+x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Dec 09 2013 *)
%t A064306 Table[FullSimplify[2^(n+1)*(2*n+2)! * Hypergeometric2F1Regularized[1, n+3/2, n+3, -8]/(n+1)! + (-1)^n/2],{n,0,20}] (* _Vaclav Kotesovec_, Dec 09 2013 *)
%t A064306 Table[(-1)^n*Sum[(-2)^k * CatalanNumber[k], {k,0,n}], {n,0,50}] (* _G. C. Greubel_, Jan 27 2017 *)
%o A064306 (Sage)
%o A064306 def A064306():
%o A064306     f, c, n = 1, 1, 1
%o A064306     while True:
%o A064306         yield f
%o A064306         n += 1
%o A064306         c = c * (8*n - 12) // n
%o A064306         f = c - f
%o A064306 a = A064306()
%o A064306 print([next(a) for _ in range(21)]) # _Peter Luschny_, Nov 30 2016
%o A064306 (PARI) for(n=0, 25, print1((-1)^n*sum(k=0,n, (-2)^k*binomial(2*k,k)/(k+1)), ", ")) \\ _G. C. Greubel_, Jan 27 2017
%K A064306 nonn,easy
%O A064306 0,3
%A A064306 _Wolfdieter Lang_, Sep 13 2001