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

0,2

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

Cf. A195761, A161809, A112765.

Sequence in context: A373964 A275935 A137360 * A195761 A100355 A048032

Adjacent sequences: A195757 A195758 A195759 * A195761 A195762 A195763

Keyword

nonn

Author

Paul D. Hanna, Sep 23 2011

Status

approved