A145347 G.f. satisfies: A(x/A(x)) = 1 + x*A(x)^3.

1, 1, 4, 26, 220, 2203, 24836, 306104, 4047988, 56713521, 834286612, 12801754120, 203889888832, 3357619794321, 56999146850380, 995081586539016, 17830012791062632, 327376145842252333, 6151225530281186372, 118142009771446643592, 2317165307900630229384

Offset

0,3

Comments

More generally, if g.f. A(x) satisfies: A(x/A(x)^k) = 1 + x*A(x)^m, then

A(x) = 1 + x*G(x)^(m+k) where G(x) = A(x*G(x)^k) and G(x/A(x)^k) = A(x);

thus a(n) = [x^(n-1)] ((m+k)/(m+k*n))*A(x)^(m+k*n) for n>=1 with a(0)=1.

Links

Table of n, a(n) for n=0..20.

Formula

G.f.: A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)) and A(x) = G(x/A(x)).

a(n) = [x^(n-1)] 4*A(x)^(n+3)/(n+3) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 4*A(x)^(n+3)/(n+3) for n>=1 (see comment).

Example

G.f.: A(x) = 1 + x + 4*x^2 + 26*x^3 + 220*x^4 + 2203*x^5 + 24836*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 103*x^3 + 876*x^4 + 8679*x^5 + 96382*x^6 +...
A(x/A(x)) = 1 + x + 3*x^2 + 15*x^3 + 103*x^4 + 876*x^5 + 8679*x^6 +...
A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)):
G(x) = 1 + x + 5*x^2 + 39*x^3 + 381*x^4 + 4284*x^5 + 53163*x^6 +...
To illustrate the formula a(n) = [x^(n-1)] 4*A(x)^(n+3)/(n+3),
form a table of coefficients in A(x)^(n+3) as follows:
A^4: [(1), 4, 22, 156, 1337, 13220, 145988, 1759876, ...];
A^5: [1, (5), 30, 220, 1905, 18836, 207100, 2481740, ...];
A^6: [1, 6, (39), 296, 2595, 25704, 281727, 3358488, ...];
A^7: [1, 7, 49, (385), 3423, 34020, 372141, 4416658, ...];
A^8: [1, 8, 60, 488, (4406), 44000, 480900, 5686480, ...];
A^9: [1, 9, 72, 606, 5562, (55881), 610872, 7202268, ...]; ...
in which the main diagonal forms the initial terms of this sequence:
[4/4*(1), 4/5*(5), 4/6*(39), 4/7*(385), 4/8*(4406), 4/9*(55881), ...].

Prog

(PARI) {a(n)=local(F=1+x); for(i=0, n, G=serreverse(x/(F+x*O(x^n))^1)/x; F=1+x*G^4); polcoeff(F, n)}
(PARI) /* This sequence is generated when k=1, m=3: A(x/A(x)^k) = 1 + x*A(x)^m */
{a(n, k=1, m=3)=local(A=sum(i=0, n-1, a(i, k, m)*x^i)+x*O(x^n)); if(n==0, 1, polcoeff((m+k)/(m+k*n)*A^(m+k*n), n-1))}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A145348, A145350, A147664, A120972.

Sequence in context: A364973 A120971 A187826 * A219780 A259902 A089816

Adjacent sequences: A145344 A145345 A145346 * A145348 A145349 A145350

Keyword

nonn

Author

Paul D. Hanna, Nov 09 2008

Status

approved