A188457 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1 + 3^n*x)^(n+1).

1, 1, 5, 109, 9449, 3068281, 3586048685, 14668583277349, 205716978569685329, 9737002299093315531121, 1536239893108209683958428885, 799846636937376803320381186364509, 1362900713950636674946135205457794784569

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.

a(n) is the number of acyclic 2-multidigraphs. Cf. A137435, A339768. - Geoffrey Critzer, Feb 21 2021

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

Cf. A003024, A137435, A188456, A188455, A135079.

Cf. A137435, A339768.

Sequence in context: A322896 A296743 A358781 * A245106 A244004 A003465

Adjacent sequences: A188454 A188455 A188456 * A188458 A188459 A188460

Keyword

nonn,easy

Author

Paul D. Hanna, Mar 31 2011

Status

approved