A145348 G.f. satisfies: A(x/A(x)^2) = 1 + x*A(x)^2.
1, 1, 4, 30, 312, 3965, 57824, 933998, 16346728, 305539046, 6037780164, 125227212342, 2711254371568, 61021656441091, 1423063422363676, 34297379607790288, 852463916004336464, 21812807282389353798
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 312*x^4 + 3965*x^5 +... A(x)^2 = 1 + 2*x + 9*x^2 + 68*x^3 + 700*x^4 + 8794*x^5 + 126974*x^6+.. A(x/A(x)^2) = 1 + x + 2*x^2 + 9*x^3 + 68*x^4 + 700*x^5 + 8794*x^6 +... A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^2): G(x) = 1 + x + 6*x^2 + 59*x^3 + 742*x^4 + 10877*x^5 + 177612*x^6 +... G(x)^2 = 1 + 2*x + 13*x^2 + 130*x^3 + 1638*x^4 + 23946*x^5 +... To illustrate the formula a(n) = [x^(n-1)] 2*A(x)^(2*n+2)/(n+1), form a table of coefficients in A(x)^(2*n+2) as follows: A^4: [(1), 4, 22, 172, 1753, 21612, 306348, ...]; A^6: [1, (6), 39, 320, 3267, 39756, 554595, ...]; A^8: [1, 8, (60), 520, 5366, 64816, 892308, ...]; A^10: [1, 10, 85, (780), 8190, 98702, 1344920, ...]; A^12: [1, 12, 114, 1108, (11895), 143676, 1943488, ...]; A^14: [1, 14, 147, 1512, 16653, (202384), 2725541, ...]; ... in which the main diagonal forms the initial terms of this sequence: [2/2*(1), 2/3*(6), 2/4*(60), 2/5*(780), 2/6*(11895), 2/7*(202384), ...].
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..230
Programs
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PARI
{a(n)=local(F=1+x,G);for(i=0,n,G=serreverse(x/(F+x*O(x^n))^2)/x;F=1+x*G^2);polcoeff(F,n)} -
PARI
/* This sequence is generated when k=2, m=2: A(x/A(x)^k) = 1 + x*A(x)^m */ {a(n, k=2, m=2)=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),", "))
Formula
G.f.: A(x) = 1 + x*G(x)^4 where G(x) = A(x*G(x)^2) and A(x) = G(x/A(x)^2).
a(n) = [x^(n-1)] 2*A(x)^(2*n+2)/(n+1) for n>=1 with a(0)=1; i.e., a(n) equals the coefficient of x^(n-1) in 2*A(x)^(2*n+2)/(n+1) for n>=1 (see comment).
Comments