OFFSET
0,3
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..62
FORMULA
G.f.: 1 = Sum_{n>=0} a(n)*C(n+m-1,n)*x^n/(1 + 3^n*x)^(n+m) for m>=1.
L.g.f.: log(1+x) = Sum_{n>=1} a(n)*(x^n/n)/(1 + 3^n*x)^n.
E.g.f.: 1 = Sum_{n>=0} a(n)*exp(-3^n*x)*x^n/n!.
a(n) = Sum_{k=1..n} (-1)^(k+1)*C(n, k)*3^(k*(n-k))*a(n-k) for n>0 with a(0)=1.
From Peter Bala, Apr 01 2013: (Start)
Let E(x) = sum {n >= 0} x^n/(n!*3^C(n,2)). Then a generating function for this sequence is 1/E(-x) = sum {n >= 0} a(n)*x^n/(n!*3^C(n,2)) = 1 + x + 5*x^2/(2!*3) + 109*x^3/(3!*3^3) + 9449*x^4/(4!*3^6) + .... (End)
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 3*x^2 + 39*x^3 + 2403*x^4 + 616131*x^5 + ... appears to have integer coefficients. - Peter Bala, Jan 14 2016
EXAMPLE
Illustration of the generating functions.
E.g.f.: 1 = exp(-x) + exp(-3*x)*x + 5*exp(-9*x)*x^2/2! + 109*exp(-27*x)*x^3/3! +...
L.g.f.: log(1+x) = x/(1+3*x) + 5*(x^2/2)/(1+9*x)^2 + 109*(x^3/3)/(1+27*x)^3 +...
G.f.: 1 = 1/(1+x) + 1*x/(1+3*x)^2 + 5*x^2/(1+9*x)^3 + 109*x^3/(1+27*x)^4 +...
G.f.: 1 = 1/(1+x)^2 + 1*2*x/(1+3*x)^3 + 5*3*x^2/(1+9*x)^4 + 109*4*x^3/(1+27*x)^5 +...
G.f.: 1 = 1/(1+x)^3 + 1*3*x/(1+3*x)^4 + 5*6*x^2/(1+9*x)^5 + 109*10*x^3/(1+27*x)^6 +...
MATHEMATICA
a[n_] := a[n] = If[n == 0, 1, Sum[-(-1)^k Binomial[n, k] 3^(k(n-k)) a[n-k], {k, 1, n}]];
a /@ Range[0, 12] (* Jean-François Alcover, Nov 02 2019 *)
PROG
(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/(1+3^k*x+x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Holds for m>=1: */
{a(n)=local(m=1); polcoeff(1-sum(k=0, n-1, a(k)*binomial(m+k-1, k)*x^k/(1+3^k*x+x*O(x^n))^(k+m)), n)/binomial(m+n-1, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* Recurrence: */
{a(n)=if(n<1, n==0, sum(k=1, n, -(-1)^k*binomial(n, k)*3^(k*(n-k))*a(n-k)))}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* E.g.f.: */
{a(n)=n!*polcoeff(1-sum(k=0, n-1, a(k)*exp(-3^k*x+x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Paul D. Hanna, Mar 31 2011
STATUS
approved