A251176 G.f. A(x) satisfies A(x) = 1 + x/(1-9*x^3)^(1/3) * A( x/sqrt(1-9*x^3)^(1/3) ).

1, 1, 1, 1, 4, 10, 19, 49, 154, 415, 1066, 3322, 10639, 31279, 96751, 326908, 1082065, 3514285, 12068659, 42713311, 148480885, 523469791, 1915394458, 7043990266, 25840482346, 96913386589, 370304345755, 1415830556098, 5458805003308, 21434092603255, 84865526136793, 337266100883830

Offset

0,5

Links

Table of n, a(n) for n=0..31.

Formula

G.f.: Sum_{n>=0} x^n / Product_{k=0..n} (1 - 9*k*x^3)^(1/3).

G.f. A(x) satisfies: A( x/sqrt(1+9*x^3)^(1/3) ) = 1 + x*A(x).

a(n) = Sum_{k=0..(n-1)/3} a(n-3*k-1) * binomial(n/3-1,k) * 9^k, for n>0 with a(0)=1. [After Vladimir Kruchinin in A201169.]

Example

G.f.: A(x) = 1 + x + x^2 + x^3 + 4*x^4 + 10*x^5 + 19*x^6 + 49*x^7 +...
such that A(x) = 1 + G(x)*A(G(x)) where
G(x) = x/(1-9*x^3)^(1/3) = x + 3*x^4 + 18*x^7 + 126*x^10 + 945*x^13 + 7371*x^16 + 58968*x^19 + 480168*x^22 +...+ A004987(n)*x^(3*n+1) +...
The g.f. also equals the series
A(x) = 1 + x/(1-9*x^3)^(1/3) + x^2/((1-9*x^3)*(1-18*x^3))^(1/3) + x^3/((1-9*x^3)*(1-18*x^3)*(1-27*x^3))^(1/3) + x^4/((1-9*x^3)*(1-18*x^3)*(1-27*x^3)*(1-36*x^3))^(1/3) +...

Prog

(PARI) {a(n)=polcoeff( sum(m=0, n, x^m / prod(k=0, m, (1 - 9*k*x^3 +x*O(x^n))^(1/3) )), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=if(n==0, 1, sum(k=0, (n-1)\3, a(n-3*k-1) * binomial(n/3-1, k) * 3^(2*k)))}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A201169, A004987.

Sequence in context: A155348 A155269 A155322 * A155428 A155383 A155229

Adjacent sequences: A251173 A251174 A251175 * A251177 A251178 A251179

Keyword

nonn

Author

Paul D. Hanna, Nov 30 2014

Status

approved