A073063 G.f. satisfies: A(x) = exp( Sum_{n>=1} L(n)*A(x^n)*x^n/n ) where L(n) = n-th Lucas number.

1, 1, 3, 7, 19, 48, 134, 362, 1026, 2915, 8463, 24760, 73439, 219444, 661592, 2007631, 6131180, 18823235, 58072904, 179931279, 559676932, 1746983911, 5470554480, 17180641614, 54101612326, 170784939844, 540351318828, 1713234349627, 5442599443734, 17321540546788

Offset

0,3

Links

Table of n, a(n) for n=0..29.

Formula

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

G.f.: A(x) = exp( Sum(n>=1} x^n/n * Sum_{k=0..n} C(n,k)*x^k*A(x^(n+k)) ). [From Paul D. Hanna, Nov 03 2011]

Example

G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 19*x^4 + 48*x^5 + 134*x^6 +...
where
log(A(x)) = A(x)*x + 3*A(x^2)*x^2/2 + 4*A(x^3)*x^3/3 + 7*A(x^4)*x^4/4 + 11*A(x^5)*x^5/5 +...
Equivalently,
log(A(x)) = (A(x)+x*A(x^2))*x + (A(x^2)+2*x*A(x^3)+x^2*A(x^4))*x^2/2 + (A(x^3)+3*x*A(x^4)+3*x^2*A(x^5)+x^3*A(x^6))*x^3/3 + (A(x^4)+4*x*A(x^5)+6*x^2*A(x^6)+4*x^3*A(x^7)+x^4*A(x^8))*x^4/4 +...

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m/m*sum(k=0, m, binomial(m, k)*x^k*subst(A, x, x^(m+k)+x*O(x^n)))))); polcoeff(A, n)}

Crossrefs

Cf. A199102, A000081, A000204 (Lucas numbers).

Sequence in context: A146810 A246493 A293733 * A370169 A007288 A191824

Adjacent sequences: A073060 A073061 A073062 * A073064 A073065 A073066

Keyword

nonn

Author

Paul D. Hanna, Aug 17 2002

Status

approved