A251687 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / A(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (1 + x + 2*x^2)^n.

1, 1, 1, 5, 8, 8, 16, 28, 48, 80, 128, 208, 320, 512, 768, 1216, 1792, 2816, 4096, 6400, 9216, 14336, 20480, 31744, 45056, 69632, 98304, 151552, 212992, 327680, 458752, 704512, 983040, 1507328, 2097152, 3211264, 4456448, 6815744, 9437184, 14417920, 19922944, 30408704, 41943040

Offset: 0

Author

Paul D. Hanna, Mar 03 2015

Keywords

• nonn

Comments

More generally, if G(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} T(n,k)^2 * x^k] / G(x)^n * x^n/n ), where T(n,k) is the coefficient of x^k in (p + q*x + r*x^2)^n, then G(x) = (1 + p^2*x)*(1 + r^2*x^3)*(1 + (q^2-2*p*r)*x^2 + p^2*r^2*x^4) / (1-p*r*x^2)^2.

Examples

			G.f.: A(x) = 1 + x + x^2 + 5*x^3 + 8*x^4 + 8*x^5 + 16*x^6 + 28*x^7 +...
where
log(A(x)) = (1 + x + 2^2*x^2)/A(x) * x +
(1 + 2^2*x + 5^2*x^2 + 4^2*x^3 + 4^2*x^4)/A(x)^2 * x^2/2 +
(1 + 3^2*x + 9^2*x^2 + 13^2*x^3 + 18^2*x^4 + 12^2*x^5 + 8^2*x^6)/A(x)^3 * x^3/3 +
(1 + 4^2*x + 14^2*x^2 + 28^2*x^3 + 49^2*x^4 + 56^2*x^5 + 56^2*x^6 + 32^2*x^7 + 16^2*x^8)/A(x)^4 * x^4/4 +
(1 + 5^2*x + 20^2*x^2 + 50^2*x^3 + 105^2*x^4 + 161^2*x^5 + 210^2*x^6 + 200^2*x^7 + 160^2*x^8 + 80^2*x^9 + 32^2*x^10)/A(x)^5 * x^5/5 +...
which involves the squares of coefficients in (1 + x + 2*x^2)^n - see triangle A084600.
		

Crossrefs

Cf. A084600, A251688, A251689, A200537.

Programs

PARI

/* By Definition: */
{a(n,p=1,q=1,r=2)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, n, polcoeff((p + q*x + r*x^2 +x*O(x^k))^m, k)^2 *x^k) *x^m/(A+x*O(x^n))^m/m)+x*O(x^n))); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

PARI

/* By G.F. Identity (faster): */
{a(n,p=1,q=1,r=2)=polcoeff( (1 + p^2*x)*(1 + r^2*x^3)*(1 + (q^2-2*p*r)*x^2 + p^2*r^2*x^4) / ((1-p*r*x^2)^2 +x*O(x^n)), n)}
for(n=0, 50, print1(a(n), ", "))

Formula

G.f.: (1 + x)*(1 + 4*x^3)*(1 - 3*x^2 + 4*x^4) / (1 - 2*x^2)^2.