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A335501 a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k). 0

%I #7 Jul 12 2020 19:24:16

%S 1,2,10,86,1098,19142,431926,12150518,414474570,16781350694,

%T 792845706630,43107783435158,2666346336398454,185796230244565462,

%U 14464057604306584774,1248919312238777955086,118855834572748011228490,12397162719421869533115622

%N a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} binomial(n,k)^2 * (n-k) * a(k).

%F a(n) = (n!)^2 * [x^n] exp(2 * Sum_{k>=1} x^k / (k!)^2).

%F a(n) = (n!)^2 * [x^n] exp(2 * (BesselI(0,2*sqrt(x)) - 1)).

%t a[0] = 1; a[n_] := a[n] = (2/n) Sum[Binomial[n, k]^2 (n - k) a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 17}]

%t nmax = 17; CoefficientList[Series[Exp[2 Sum[x^k/(k!)^2, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2

%Y Cf. A001861, A023998.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jul 12 2020

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