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A356280
a(n) = Sum_{k=0..n} binomial(2*n, n-k) * p(k), where p(k) is the partition function A000041.
4
1, 3, 12, 50, 211, 894, 3791, 16068, 68032, 287675, 1214761, 5122428, 21571028, 90718913, 381050570, 1598645263, 6699355413, 28044720813, 117281866330, 489999068614, 2045341248508, 8530263939665, 35547083083270, 148015639243691, 615870619714675, 2560734764460360
OFFSET
0,2
LINKS
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, VI.26. Catalan sums, p.417.
FORMULA
a(n) ~ 2^(2*n - 1/2) * exp(3^(1/3) * Pi^(4/3) * n^(1/3) / 4) / (3*Pi*n)^(2/3).
MATHEMATICA
Table[Sum[PartitionsP[k]*Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[PartitionsP[k]*((1-2*x-Sqrt[1-4*x])/(2*x))^k / Sqrt[1-4*x], {k, 0, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Aug 01 2022
STATUS
approved