A274272 - OEIS

%I #12 Jun 23 2016 10:05:23

%S 1,6,185,15246,1736385,212946246,26516391385,3312004971246,

%T 413937039016385,51740540399346246,6467527813385891385,

%U 808439983261977471246,101054973072475964016385,12631871013177766274346246,1578983861125177809948391385

%N Number of partitions of 5^n into at most four parts.

%H Colin Barker, <a href="/A274272/b274272.txt">Table of n, a(n) for n = 0..450</a>

%F Coefficient of x^(5^n) in 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).

%F Conjectures (Start)

%F a(n) = (57+8*(-1)^n+63*5^n+3*5^(1+2*n)+125^n)/144.

%F a(n) = 155*a(n-1)-3874*a(n-2)+15470*a(n-3)+3875*a(n-4)-15625*a(n-5) for n>4.

%F G.f.: (1-149*x+3129*x^2-5655*x^3-6750*x^4) / ((1-x)*(1+x)*(1-5*x)*(1-25*x)*(1-125*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)*(1-x^4)).

%o b(n) = round(real(68+36*(-1)^n+18*((-I)^n+I^n)+(16*exp(-2/3*I*n*Pi)*(1+I*sqrt(3)+2*exp((4*I*n*Pi)/3)))/(1+(-1)^(1/3))+59*(1+n)+9*(-1)^n*(1+n)+18*(1+n)*(2+n)+2*(1+n)*(2+n)*(3+n))/288)

%o vector(20, n, n--; b(5^n))

%Y A subsequence of A001400.

%Y Cf. A274100 (2^n), A274271 (3^n).

%K nonn

%O 0,2

%A _Colin Barker_, Jun 17 2016