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

0,2

Links

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

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

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 *)

Prog

(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

Crossrefs

Third column (m=2) of triangle A112705.

Sequence in context: A113283 A326519 A127784 * A346967 A192925 A350354

Adjacent sequences: A112693 A112694 A112695 * A112697 A112698 A112699

Keyword

nonn,easy

Author

Wolfdieter Lang, Oct 31 2005

Status

approved