A210097 G.f.: 1 = Sum_{n>=0} a(n)*x^n*(1-x)^(n+1)/(1 + (n+1)*x)^(2*n+1).

1, 2, 14, 180, 3464, 90018, 2968968, 119252060, 5664108920, 311212995732, 19447137310280, 1363465685120296, 106080538006819320, 9075286815352325570, 847179165282912257160, 85729124193701299050252, 9351018056855217246637304, 1094013446495887840421430300

Offset

0,2

Comments

Compare to a g.f. for A000272: 1 = Sum_{n>=0} (n+1)^(n-1) * x^n/(1 + (n+1)*x)^(n+1).

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = Sum_{k=1..n} -(-1)^k*binomial(2*n-k,k)*a(n-k)*(n-k+2)^k for n>=1 with a(0)=1.

Example

G.f.: 1 = 1*(1-x)/(1+x) + 2*x*(1-x)^2/(1+2*x)^3 + 14*x^2*(1-x)^3/(1+3*x)^5 + 180*x^3*(1-x)^4/(1+4*x)^7 + 3464*x^4*(1-x)^5/(1+5*x)^9 + 90018*x^5*(1-x)^6/(1+6*x)^11 +...
Compare to a g.f. for A000272:
1 = 1/(1+x) + x/(1+2*x)^2 + 3*x^2/(1+3*x)^3 + 4^2*x^3/(1+4*x)^4 + 5^3*x^4/(1+5*x)^5 + 6^4*x^5/(1+6*x)^6 + 7^5*x^6/(1+7*x)^7 +...

Prog

(PARI) {a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k*(1-x)^(k+1)/(1+(k+1)*x+x*O(x^n))^(2*k+1)), n)}
(PARI) {a(n)=if(n==0, 1, sum(k=1, n, -(-1)^k*binomial(2*n-k, k)*a(n-k)*(n-k+2)^k))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A210096, A000272.

Sequence in context: A285270 A109520 A370054 * A230991 A258872 A000807

Adjacent sequences: A210094 A210095 A210096 * A210098 A210099 A210100

Keyword

nonn

Author

Paul D. Hanna, Mar 17 2012

Status

approved