login
A309682
G.f.: C(x)*C(2*x^2)*C(3*x^3)*..., where C(x) is the g.f. for A000108.
2
1, 1, 4, 10, 33, 81, 282, 762, 2599, 7979, 27343, 89371, 315256, 1078498, 3857048, 13651786, 49475282, 178736186, 655247192, 2401663838, 8883371016, 32906649488, 122619768860, 457836275272, 1716620421629, 6449729802639, 24308647131627, 91800114425437
OFFSET
0,3
LINKS
FORMULA
a(n) ~ c * 4^n / n^(3/2), where c = 1/(2*sqrt(Pi)) * Product_{k>=1} (2^k*(2^(k-1) - sqrt(4^(k-1) - k))/k) = 0.711438694828613555153724789...
MAPLE
C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1,
C(n), add(C(j)*i^j*b(n-i*j, i-1), j=0..n/i))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30); # Alois P. Heinz, Aug 23 2019
MATHEMATICA
nmax = 30; CoefficientList[Series[Product[Sum[CatalanNumber[k]*j^k*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*k*x^k])/(2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Aug 12 2019
STATUS
approved