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

1, 3, 11, 51, 275, 1619, 10067, 64979, 431059, 2920403, 20119507, 140513235, 992530387, 7078367187, 50896392147, 368577073107, 2685777334227, 19678579249107, 144888698621907, 1071443581980627, 7954422715502547

Offset: 0

Author

Wolfdieter Lang, Oct 31 2005

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Third column (m=2) of triangle A112705.

Programs

Maple

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

Mathematica

Table[Sum[Binomial[2*j,j]*2^j/(j+1),{j,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 19 2012 *)

Sage

def A112696():
    f, c, n = 1, 1, 1
    while True:
        yield f
        n += 1
        c = c * (8*n - 12) // n
        f += c
a = A112696()
print([next(a) for  in range(21)]) # _Peter Luschny, Nov 30 2016

Formula

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

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.

a(n) = Sum_{j=0..n} binomial(2*j,j)*2^j/(j+1). - Zerinvary Lajos, Oct 26 2006

Recurrence: (n+1)*a(n) = 3*(3*n-1)*a(n-1) - 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 2^(3*n+3)/(7*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012