OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: A(x) = 1/(1-x)^5 * Product_{n>=1} 1/(1 - x^(5^n))^4.
G.f. satisfies: A(x) = A(x^5)*(1-x^5)/(1-x)^5.
Let the QUINTISECTIONS of g.f. A(x) be defined by:
A(x) = Q0(x^5) + x*Q1(x^5) + x^2*Q3(x^5) + x^3*Q3(x^5) + x^4*Q4(x^5),
then:
_ Q0(x) = (1 + 121*x + 381*x^2 + 121*x^3 + x^4)/R(x)
_ Q1(x) = 5*(1 + 37*x + 73*x^2 + 14*x^3)/R(x)
_ Q2(x) = 5*(3 + 51*x + 64*x^2 + 7*x^3)/R(x)
_ Q3(x) = 5*(7 + 64*x + 51*x^2 + 3*x^3)/R(x)
_ Q4(x) = 5*(14 + 73*x + 37*x^2 + 1*x^3)/R(x)
where R(x) = (1-x)^5 * Product_{n>=0} (1 - x^(5^n))^4.
Further, the quintisections are related by:
_ Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2.
EXAMPLE
G.f.: A(x) = 1 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 130*x^5 + 230*x^6 +...
log(A(x)) = 5*x + 5*x^2/2 + 5*x^3/3 + 5*x^4/4 + 25*x^5/5 + 5*x^6/6 + 5*x^7/7 + 5*x^8/8 + 5*x^9/9 + 25*x^10/10 +...
The coefficients in the QUINTISECTIONS of g.f. A(x) begin:
Q0: [1, 130, 1515, 9160, 38695, 129505, 367855, 926265, 2128510, ...];
Q1: [5, 230, 2255, 12505, 50055, 161405, 446255, 1101155, 2491030, ...];
Q2: [15, 390, 3290, 16865, 64215, 199965, 538965, 1304615, 2907440, ...];
Q3: [35, 635, 4710, 22485, 81735, 246335, 648185, 1540635, 3384660, ...];
Q4: [70, 995, 6620, 29645, 103245, 301795, 776345, 1813595, 3930245, ...].
The coefficients in the products Q2*Q3 and Q1*Q4 begin:
Q2(x)*Q3(x): [525, 23175, 433450, 4853600, 38447875, 236756775, ...];
Q1(x)*Q4(x): [350, 21075, 419800, 4789900, 38209000, 235990975, ...];
where Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2, and
R(x) = 1 - 9*x + 36*x^2 - 84*x^3 + 126*x^4 - 130*x^5 + 120*x^6 +...
PROG
(PARI) {a(n)=local(N=ceil(log(n+6)/log(5))); polcoeff(1/(1-x+x*O(x^n))^5/prod(k=1, N, (1-x^(5^k) +x*O(x^n))^4), n)}
(PARI) {a(n)=local(L=sum(m=1, n, 5*5^valuation(m, 5)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 23 2011
STATUS
approved