|
| |
|
|
A188457
|
|
G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1 + 3^n*x)^(n+1).
|
|
5
|
|
|
|
1, 1, 5, 109, 9449, 3068281, 3586048685, 14668583277349, 205716978569685329, 9737002299093315531121, 1536239893108209683958428885, 799846636937376803320381186364509, 1362900713950636674946135205457794784569
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
G.f. satisfies a variant of an identity involving A003024:
1 = Sum_{n>=0} A003024(n)*x^n/(1 + 2^n*x)^(n+1),
where A003024(n) is the number of acyclic digraphs with n labeled nodes.
|
|
|
LINKS
|
Table of n, a(n) for n=0..12.
|
|
|
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) + ....
Recurrence equation: a(n) = sum {k = 0..n-1} C(n,k)*(-1)^(n-k+1)*3^(k*(n-k))*a(k) with a(0) = 1. (End)
|
|
|
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 +...
|
|
|
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)}
(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)}
(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)))}
(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)}
|
|
|
CROSSREFS
|
Cf. A003024, A137435, A188456, A188455. A135079.
Sequence in context: A014180 A012122 A012091 * A003465 A177680 A215786
Adjacent sequences: A188454 A188455 A188456 * A188458 A188459 A188460
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
Paul D. Hanna, Mar 31 2011
|
|
|
STATUS
|
approved
|
| |
|
|
