A195761 - OEIS

%I #7 Mar 30 2012 18:37:29

%S 1,5,15,35,70,135,255,465,810,1345,2180,3480,5465,8410,12645,18720,

%T 27405,39690,56785,80120,111840,154805,212590,289485,390495,522640,

%U 694955,918490,1206310,1573495,2040260,2631955,3379065,4317210,5487145,6941780,8746180,10977565,13725310

%N G.f.: exp( Sum_{n>=1} A055457(n) * 5^A055457(n) * x^n/n ) where 5^A055457(n) exactly divides 5*n.

%H Paul D. Hanna, <a href="/A195761/b195761.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: A(x) = Product_{n>=0} 1/(1 - x^(5^n))^(4*n+5).

%F G.f. satisfies: A(x) = (1-x^5)/(1-x)^5 * A(x^5)^2/A(x^25).

%F G.f. satisfies: A(x) = A(x^5)*G(x) where G(x) = G(x^5)*(1-x^5)/(1-x)^5 is the g.f. of A195760.

%F Let the QUINTISECTIONS of A(x) be defined by:

%F 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 then

%F _ Q0(x) = (1 + 121*x + 381*x^2 + 121*x^3 + x^4)/R(x)

%F _ Q1(x) = 5*(1 + 37*x + 73*x^2 + 14*x^3)/R(x)

%F _ Q2(x) = 5*(3 + 51*x + 64*x^2 + 7*x^3)/R(x)

%F _ Q3(x) = 5*(7 + 64*x + 51*x^2 + 3*x^3)/R(x)

%F _ Q4(x) = 5*(14 + 73*x + 37*x^2 + 1*x^3)/R(x)

%F where R(x) = (1-x)^5*Product_{n>=0} (1 - x^(5^n))^(4*n+9).

%F Further, the quintisections are related by:

%F _ Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2.

%e G.f.: A(x) = 1 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 135*x^5 + 255*x^6 +...

%e log(A(x)) = 5*x + 5*x^2/2 + 5*x^3/3 + 5*x^4/4 + 50*x^5/5 + 5*x^6/6 + 5*x^7/7 + 5*x^8/8 + 5*x^9/9 + 50*x^10/10 +...

%e The coefficients in the QUINTISECTIONS of g.f. A(x) begin:

%e Q0: [1, 135, 2180, 18720, 111840, 522640, 2040260, 6941780, ...];

%e Q1: [5, 255, 3480, 27405, 154805, 694955, 2631955, 8746180, ...];

%e Q2: [15, 465, 5465, 39690, 212590, 918490, 3379065, 10977565, ...];

%e Q3: [35, 810, 8410, 56785, 289485, 1206310, 4317210, 13725310, ...];

%e Q4: [70, 1345, 12645, 80120, 390495, 1573495, 5487145, 17090945, ...].

%o (PARI) {a(n)=local(N=ceil(log(n+6)/log(5)));polcoeff(1/prod(k=0,N,(1-x^(5^k) +x*O(x^n))^(4*k+5)),n)}

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, valuation(5*m, 5)*5^valuation(5*m, 5)*x^m/m)+x*O(x^n)), n)}

%Y Cf. A195760, A183038, A183036, A055457.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 23 2011