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a(n) = n! * Sum_{k=0..n} (2*k)! / (k!)^3.
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%I #9 Nov 20 2021 20:58:02

%S 1,3,12,56,294,1722,11256,82224,670662,6084578,61030536,672041328,

%T 8067200092,104884001796,1468416141744,22026397243680,352422956979270,

%U 5991192602253810,107841475915703880,2048988077743637520,40979761692719279220,860574996085362738060

%N a(n) = n! * Sum_{k=0..n} (2*k)! / (k!)^3.

%F E.g.f.: exp(2*x) * BesselI(0,2*x) / (1 - x).

%F a(n) = Sum_{k=0..n} binomial(n,k) * A000984(k) * A000142(n-k).

%F a(n) = Sum_{k=0..n} binomial(n,k) * A002426(k) * A000522(n-k).

%F a(n) ~ exp(2) * BesselI(0,2) * n!. - _Vaclav Kotesovec_, Nov 20 2021

%t Table[n! Sum[(2 k)!/(k!)^3, {k, 0, n}], {n, 0, 21}]

%t nmax = 21; CoefficientList[Series[Exp[2 x] BesselI[0, 2 x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

%o (PARI) a(n) = n! * sum(k=0, n, (2*k)! / (k!)^3) \\ _Andrew Howroyd_, Nov 20 2021

%Y Cf. A000142, A000522, A000984, A002426, A026375, A052143, A336293.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Nov 20 2021