A199257 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} C(2*n,k)^2 * x^k * A(x)^k]* x^n/n ).
1, 1, 5, 18, 86, 408, 2075, 10787, 57655, 313643, 1733450, 9700574, 54867895, 313145033, 1801150861, 10430094658, 60758092753, 355795743385, 2093295146379, 12367548160650, 73346850194969, 436486017193373, 2605656191324094, 15599323024019360, 93634195155551584
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 5*x^2 + 18*x^3 + 86*x^4 + 408*x^5 + 2075*x^6 +... such that A(x) = G(x*A(x)) where G(x) = (1-x+x^2)*(1+x^2)^2/(1-x)^2: G(x) = 1 + x + 4*x^2 + 5*x^3 + 9*x^4 + 12*x^5 + 16*x^6 + 20*x^7 + 24*x^8 +... ... Let A = x*A(x), then the logarithm of the g.f. A(x) equals the series: log(A(x)) = (1 + 2^2*A + A^2)*x + (1 + 4^2*A + 6^2*A^2 + 4^2*A^3 + A^4)*x^2/2 + (1 + 6^2*A + 15^2*A^2 + 20^2*A^3 + 15^2*A^4 + 6^2*A^5 + A^6)*x^3/3 + (1 + 8^2*A + 28^2*A^2 + 56^2*A^3 + 70^2*A^4 + 56^2*A^5 + 28^2*A^6 + 8^2*A^7 + A^8)*x^4/4 + (1 + 10^2*A + 45^2*A^2 + 120^2*A^3 + 210^2*A^4 + 252^2*A^5 + 210^2*A^6 + 120^2*A^7 + 45^2*A^8 + 10^2*A^9 + A^10)*x^5/5 +... which involves the squares of binomial coefficients C(2*n,k).
Programs
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PARI
{a(n)=local(A=1+x); A=1/x*serreverse(x*(1-x)^2/((1-x+x^2)*(1+x^2)^2+x*O(x^n))); polcoeff(A, n)} -
PARI
{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, binomial(2*m, k)^2 *x^k*A^k) *x^m/m)+x*O(x^n))); polcoeff(A, n)}
Formula
G.f. satisfies: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = (1-x+x^2)*(1+x^2)^2/(1-x)^2.
G.f.: A(x) = (1/x)*Series_Reversion( x*(1-x)^2/((1-x+x^2)*(1+x^2)^2) ).