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.

1, 5, 15, 35, 70, 130, 230, 390, 635, 995, 1515, 2255, 3290, 4710, 6620, 9160, 12505, 16865, 22485, 29645, 38695, 50055, 64215, 81735, 103245, 129505, 161405, 199965, 246335, 301795, 367855, 446255, 538965, 648185, 776345, 926265, 1101155, 1304615, 1540635

Offset: 0

Author

Paul D. Hanna, Sep 23 2011

Keywords

• nonn

Examples

			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 +...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A195761, A161809, A112765.

Programs

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)}

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.