A356280 a(n) = Sum_{k=0..n} binomial(2*n, n-k) * p(k), where p(k) is the partition function A000041.

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

Author

Vaclav Kotesovec, Aug 01 2022

Keywords

• nonn

Links

• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, VI.26. Catalan sums, p.417.

Crossrefs

Cf. A000041, A032443, A218481, A286955, A356267, A356281.

Programs

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]

Formula

a(n) ~ 2^(2*n - 1/2) * exp(3^(1/3) * Pi^(4/3) * n^(1/3) / 4) / (3*Pi*n)^(2/3).