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E.g.f.: exp((3/2) * x * (2 + x)).
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%I #20 Aug 09 2021 06:34:47

%S 1,3,12,54,270,1458,8424,51516,331452,2230740,15641424,113846472,

%T 857706408,6671592216,53465326560,440602852752,3727748253456,

%U 32332181692464,287111706003648,2607272929404000,24187186030419936,228997933855499808,2210786521482955392,21746223198911853504

%N E.g.f.: exp((3/2) * x * (2 + x)).

%F G.f.: 1 / (1 - 3*x - 3*x^2 / (1 - 3*x - 6*x^2 / (1 - 3*x - 9*x^2 / (1 - 3*x - 12*x^2 / (1 - ...))))), a continued fraction.

%F D-finite with recurrence a(n) = 3 * (a(n-1) + (n-1) * a(n-2)).

%F a(n) = Sum_{k=0..n} binomial(n,k) * A000085(k) * A000898(n-k).

%F a(n) = Sum_{k=0..n} binomial(n,k) * A202830(k).

%F a(n) ~ 3^(n/2) * exp(-3/4 + sqrt(3*n) - n/2) * n^(n/2) / sqrt(2). - _Vaclav Kotesovec_, Aug 09 2021

%t nmax = 23; CoefficientList[Series[Exp[(3/2) x (2 + x)], {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[1] = 3; a[n_] := a[n] = 3 (a[n - 1] + (n - 1) a[n - 2]); Table[a[n], {n, 0, 23}]

%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp((3*x*(2 + x)/2)))) \\ _Michel Marcus_, Nov 21 2020

%Y Cf. A000085, A000898, A202830, A294119.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Nov 20 2020