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A346174 Inverse binomial transform of A317614.

%I A346174 #12 Feb 16 2025 08:34:02
%S A346174 0,1,6,30,120,420,1344,4032,11520,31680,84480,219648,559104,1397760,
%T A346174 3440640,8355840,20054016,47628288,112066560,261488640,605552640,
%U A346174 1392771072,3183476736,7235174400,16357785600,36805017600,82443239424,183911841792,408692981760,904963031040
%N A346174 Inverse binomial transform of A317614.
%H A346174 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A346174 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinomialTransform.html">Binomial Transform</a>
%H A346174 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F A346174 O.g.f.: x*(1 - 2*x + 6*x^2 - 8*x^3 + 4*x^4)/(1 - 2*x)^4.
%F A346174 E.g.f.: x*(1 + exp(2*x)*(3 + 6*x + 2*x^2))/4.
%F A346174 a(n) = 8*a(n-1) - 24*a(n-2) + 32*a(n-3) - 16*a(n-4) for n > 5.
%F A346174 a(n) = 2^(n-4)*n*(n + 1)*(n + 2) with a(0) = 0 and a(1) = 1.
%F A346174 a(n) = A000079(n-4)*A007531(n+2) for n > 1.
%F A346174 a(n) ~ A128789(n)/16.
%F A346174 Sum_{n>0} 1/a(n) = 8*log(2) - 13/3 = 1.21184411114622914200452363833...
%t A346174 LinearRecurrence[{8,-24,32,-16},{0,1,6,30,120,420},30]
%Y A346174 Cf. A000079, A007531, A128789, A257872 (-8*log(2)), A317614.
%K A346174 nonn,easy
%O A346174 0,3
%A A346174 _Stefano Spezia_, Jul 08 2021