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Expansion of e.g.f. 1 / (2 - exp(4*x)).
7

%I #5 Oct 06 2019 10:05:14

%S 1,4,48,832,19200,553984,19181568,774848512,35771842560,1857882947584,

%T 107214340620288,6805814291464192,471298297319915520,

%U 35356865248765149184,2856513752723261227008,247264693517100223823872,22830563015939200206766080,2239752722978295095737974784

%N Expansion of e.g.f. 1 / (2 - exp(4*x)).

%F a(0) = 1; a(n) = Sum_{k=1..n} 4^k * binomial(n,k) * a(n-k).

%F a(n) = Sum_{k>=0} (4*k)^n / 2^(k + 1).

%F a(n) = 4^n * A000670(n).

%p a:= proc(n) option remember; `if`(n=0, 1, add(

%p a(n-j)*binomial(n, j)*4^j, j=1..n))

%p end:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Oct 06 2019

%t nmax = 17; CoefficientList[Series[1/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!

%t a[0] = 1; a[n_] := a[n] = Sum[4^k Binomial[n, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

%t Table[2^(2 n - 1) HurwitzLerchPhi[1/2, -n, 0], {n, 0, 17}]

%Y Cf. A000670, A216794, A328182.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Oct 06 2019