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A112696 Partial sum of Catalan numbers A000108 multiplied by powers of 2.

%I A112696 #19 Feb 26 2020 06:42:01
%S A112696 1,3,11,51,275,1619,10067,64979,431059,2920403,20119507,140513235,
%T A112696 992530387,7078367187,50896392147,368577073107,2685777334227,
%U A112696 19678579249107,144888698621907,1071443581980627,7954422715502547
%N A112696 Partial sum of Catalan numbers A000108 multiplied by powers of 2.
%H A112696 Vincenzo Librandi, <a href="/A112696/b112696.txt">Table of n, a(n) for n = 0..300</a>
%F A112696 a(n) = Sum_{k=0..n} C(k)*2^k, n >= 0, with C(n):=A000108(n).
%F A112696 G.f.: c(2*x)/(1-x), where c(x):=(1-sqrt(1-4*x))/(2*x) is the o.g.f. of Catalan numbers A000108.
%F A112696 a(n) = Sum_{j=0..n} binomial(2*j,j)*2^j/(j+1). - _Zerinvary Lajos_, Oct 26 2006
%F A112696 Recurrence: (n+1)*a(n) = 3*(3*n-1)*a(n-1) - 4*(2*n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 19 2012
%F A112696 a(n) ~ 2^(3*n+3)/(7*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 19 2012
%p A112696 a:=n->sum((binomial(2*j,j))*2^j/(j+1),j=0..n): seq(a(n), n=0..20); # _Zerinvary Lajos_, Oct 26 2006
%t A112696 Table[Sum[Binomial[2*j,j]*2^j/(j+1),{j,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 19 2012 *)
%o A112696 (Sage)
%o A112696 def A112696():
%o A112696     f, c, n = 1, 1, 1
%o A112696     while True:
%o A112696         yield f
%o A112696         n += 1
%o A112696         c = c * (8*n - 12) // n
%o A112696         f += c
%o A112696 a = A112696()
%o A112696 print([next(a) for _ in range(21)]) # _Peter Luschny_, Nov 30 2016
%Y A112696 Third column (m=2) of triangle A112705.
%K A112696 nonn,easy
%O A112696 0,2
%A A112696 _Wolfdieter Lang_, Oct 31 2005