A318264 Expansion of Product_{k>=1} (1 + C(k)*x^k), where C(k) is the Catalan number A000108.

1, 1, 2, 7, 19, 66, 212, 743, 2487, 9012, 31177, 113775, 404584, 1490726, 5376676, 20028981, 73068861, 273659672, 1009921813, 3801386137, 14125670266, 53477758556, 199950414035, 759566205693, 2857261603610, 10889590477287, 41136917417501, 157329747348492

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1650

Formula

a(n) ~ c * A000108(n) ~ c * 4^n / (sqrt(Pi) * n^(3/2)), where c = Product_{k>=1} (1 + C(k)/4^k) = 2.608465265690846547082817204714986077801494... - Vaclav Kotesovec, Aug 24 2018

Maple

C:= proc(n) option remember; binomial(n+n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
     `if`(n=0, 1, b(n, i-1)+C(i)*b(n-i, min(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2019

Mathematica

nmax = 40; CoefficientList[Series[Product[1+CatalanNumber[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += CatalanNumber[k]*poly[[j - k + 1]], {j, nmax, k, -1}]; , {k, 2, nmax}]; poly

Crossrefs

Cf. A000108, A022629, A179381, A318248, A318263.

Sequence in context: A275289 A151430 A083309 * A164979 A243279 A362097

Adjacent sequences: A318261 A318262 A318263 * A318265 A318266 A318267

Keyword

nonn

Author

Vaclav Kotesovec, Aug 22 2018

Status

approved