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A195760 G.f.: A(x) = exp( Sum_{n>=1} 5*5^A112765(n) * x^n/n ), where A112765 is the exponent of the highest power of 5 dividing n.

%I A195760 #9 Mar 30 2012 18:37:29
%S A195760 1,5,15,35,70,130,230,390,635,995,1515,2255,3290,4710,6620,9160,12505,
%T A195760 16865,22485,29645,38695,50055,64215,81735,103245,129505,161405,
%U A195760 199965,246335,301795,367855,446255,538965,648185,776345,926265,1101155,1304615,1540635
%N A195760 G.f.: A(x) = exp( Sum_{n>=1} 5*5^A112765(n) * x^n/n ), where A112765 is the exponent of the highest power of 5 dividing n.
%H A195760 Paul D. Hanna, <a href="/A195760/b195760.txt">Table of n, a(n) for n = 0..1000</a>
%F A195760 G.f.: A(x) = 1/(1-x)^5 * Product_{n>=1} 1/(1 - x^(5^n))^4.
%F A195760 G.f. satisfies: A(x) = A(x^5)*(1-x^5)/(1-x)^5.
%F A195760 Let the QUINTISECTIONS of g.f. A(x) be defined by:
%F A195760 A(x) = Q0(x^5) + x*Q1(x^5) + x^2*Q3(x^5) + x^3*Q3(x^5) + x^4*Q4(x^5),
%F A195760 then:
%F A195760 _ Q0(x) = (1 + 121*x + 381*x^2 + 121*x^3 + x^4)/R(x)
%F A195760 _ Q1(x) = 5*(1 + 37*x + 73*x^2 + 14*x^3)/R(x)
%F A195760 _ Q2(x) = 5*(3 + 51*x + 64*x^2 + 7*x^3)/R(x)
%F A195760 _ Q3(x) = 5*(7 + 64*x + 51*x^2 + 3*x^3)/R(x)
%F A195760 _ Q4(x) = 5*(14 + 73*x + 37*x^2 + 1*x^3)/R(x)
%F A195760 where R(x) = (1-x)^5 * Product_{n>=0} (1 - x^(5^n))^4.
%F A195760 Further, the quintisections are related by:
%F A195760 _ Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2.
%e A195760 G.f.: A(x) = 1 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 130*x^5 + 230*x^6 +...
%e A195760 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 +...
%e A195760 The coefficients in the QUINTISECTIONS of g.f. A(x) begin:
%e A195760 Q0: [1, 130, 1515, 9160, 38695, 129505, 367855, 926265, 2128510, ...];
%e A195760 Q1: [5, 230, 2255, 12505, 50055, 161405, 446255, 1101155, 2491030, ...];
%e A195760 Q2: [15, 390, 3290, 16865, 64215, 199965, 538965, 1304615, 2907440, ...];
%e A195760 Q3: [35, 635, 4710, 22485, 81735, 246335, 648185, 1540635, 3384660, ...];
%e A195760 Q4: [70, 995, 6620, 29645, 103245, 301795, 776345, 1813595, 3930245, ...].
%e A195760 The coefficients in the products Q2*Q3 and Q1*Q4 begin:
%e A195760 Q2(x)*Q3(x): [525, 23175, 433450, 4853600, 38447875, 236756775, ...];
%e A195760 Q1(x)*Q4(x): [350, 21075, 419800, 4789900, 38209000, 235990975, ...];
%e A195760 where Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2, and
%e A195760 R(x) = 1 - 9*x + 36*x^2 - 84*x^3 + 126*x^4 - 130*x^5 + 120*x^6 +...
%o A195760 (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)}
%o A195760 (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)}
%Y A195760 Cf. A195761, A161809, A112765.
%K A195760 nonn
%O A195760 0,2
%A A195760 _Paul D. Hanna_, Sep 23 2011