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%I #10 Sep 01 2020 03:35:10
%S 1,1,7,45,439,5157,73455,1217101,23066311,491680437,11645898655,
%T 303422639517,8624098330359,265546702327813,8805478883825359,
%U 312844282877905389,11855836533424581415,477380986427269453653,20352680600044759742463,915923521948522369041469
%N a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (4*k-3) * a(n-k).
%F E.g.f.: 1 / (exp(x) * (3 - 4*x) - 2).
%F a(n) ~ n! * c * 2^(2*n+1) / ((1-c) * (3 - 4*c)^(n+1)), where c = -LambertW(-exp(-3/4)/2). - _Vaclav Kotesovec_, Aug 31 2020
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (4 k - 3) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
%t nmax = 19; CoefficientList[Series[1/(Exp[x] (3 - 4 x) - 2), {x, 0, nmax}], x] Range[0, nmax]!
%o (PARI) seq(n)={Vec(serlaplace(1 / (exp(x + O(x*x^n)) * (3 - 4*x) - 2)))} \\ _Andrew Howroyd_, Aug 31 2020
%Y Cf. A000354, A001907, A337552, A337554.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 31 2020
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