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Number of partitions of 2^n into at most three parts.
2

%I #13 Jun 16 2016 03:14:30

%S 1,2,4,10,30,102,374,1430,5590,22102,87894,350550,1400150,5596502,

%T 22377814,89494870,357946710,1431721302,5726754134,22906754390,

%U 91626493270,366504924502,1466017600854,5864066209110,23456256447830,93825009014102,375300002501974

%N Number of partitions of 2^n into at most three parts.

%H Colin Barker, <a href="/A274232/b274232.txt">Table of n, a(n) for n = 0..1000</a>

%F Coefficient of x^(2^n) in 1/((1-x)*(1-x^2)*(1-x^3)).

%F Conjectures: (Start)

%F a(n) = (8+3*2^(1+n)+4^n)/12 for n>0.

%F a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.

%F G.f.: (1-5*x+4*x^2+2*x^3) / ((1-x)*(1-2*x)*(1-4*x)).

%F (End)

%o (PARI)

%o \\ b(n) is the coefficient of x^n in the g.f. 1/((1-x)*(1-x^2)*(1-x^3)).

%o b(n) = round(real((47+9*(-1)^n + 8*exp(-2/3*I*n*Pi) + 8*exp((2*I*n*Pi)/3) + 36*n+6*n^2)/72))

%o vector(50, n, n--; b(2^n))

%Y A subsequence of A001399. Cf. A274100, A274233.

%K nonn

%O 0,2

%A _Colin Barker_, Jun 15 2016