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A318264 Expansion of Product_{k>=1} (1 + C(k)*x^k), where C(k) is the Catalan number A000108. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
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
Sequence in context: A275289 A151430 A083309 * A164979 A243279 A362097
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Aug 22 2018
STATUS
approved

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Last modified May 25 01:50 EDT 2024. Contains 372782 sequences. (Running on oeis4.)