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Partial sum of Catalan numbers A000108 multiplied by powers of 2.
3

%I #19 Feb 26 2020 06:42:01

%S 1,3,11,51,275,1619,10067,64979,431059,2920403,20119507,140513235,

%T 992530387,7078367187,50896392147,368577073107,2685777334227,

%U 19678579249107,144888698621907,1071443581980627,7954422715502547

%N Partial sum of Catalan numbers A000108 multiplied by powers of 2.

%H Vincenzo Librandi, <a href="/A112696/b112696.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = Sum_{k=0..n} C(k)*2^k, n >= 0, with C(n):=A000108(n).

%F 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 a(n) = Sum_{j=0..n} binomial(2*j,j)*2^j/(j+1). - _Zerinvary Lajos_, Oct 26 2006

%F 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 a(n) ~ 2^(3*n+3)/(7*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 19 2012

%p 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 Table[Sum[Binomial[2*j,j]*2^j/(j+1),{j,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 19 2012 *)

%o (Sage)

%o def A112696():

%o f, c, n = 1, 1, 1

%o while True:

%o yield f

%o n += 1

%o c = c * (8*n - 12) // n

%o f += c

%o a = A112696()

%o print([next(a) for _ in range(21)]) # _Peter Luschny_, Nov 30 2016

%Y Third column (m=2) of triangle A112705.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Oct 31 2005