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
Keywords
Examples
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), ...].
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..300
Programs
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PARI
{a(n) = my(F=1+x); for(i=0,n, G=serreverse(x/(F+x*O(x^n))^1)/x; F=1+x*G^4); polcoef(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) = my(A=sum(i=0, n-1, a(i, k, m)*x^i) +x*O(x^n)); if(n==0, 1, polcoef((m+k)/(m+k*n)*A^(m+k*n), n-1))} for(n=0,20,print1(a(n),", "))
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).
Comments