A199248 G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A027907(n,k)^2 * x^k * A(x)^k]* x^n/n ), where A027907 is the triangle of trinomial coefficients.
1, 1, 2, 6, 20, 69, 248, 923, 3523, 13706, 54152, 216710, 876607, 3578405, 14722432, 60986158, 254145337, 1064712328, 4481577078, 18943753140, 80381689202, 342254333393, 1461864544896, 6262021627055, 26894816382199, 115792035533779, 499648608539714, 2160504474956390
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 69*x^5 + 248*x^6 + 923*x^7 +... such that A(x) = G(x*A(x)) where G(x) is given by: G(x) = (1 - x + x^2)*(1 - x^2 + x^4)/(1-x)^2 = (1-x^5)/(1-x) + x^3/(1-x)^2: G(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 7*x^9 +... ... Let A = x*A(x), then the logarithm of the g.f. A(x) equals the series: log(A(x)) = (1 + A + A^2)*x + (1 + 2^2*A + 3^2*A^2 + 2^2*A^3 + A^4)*x^2/2 + (1 + 3^2*A + 6^2*A^2 + 7^2*A^3 + 6^2*A^4 + 3^2*A^5 + A^6)*x^3/3 + (1 + 4^2*A + 10^2*A^2 + 16^2*A^3 + 19^2*A^4 + 16^2*A^5 + 10^2*A^6 + 4^2*A^7 + A^8)*x^4/4 + (1 + 5^2*A + 15^2*A^2 + 30^2*A^3 + 45^2*A^4 + 51^2*A^5 + 45^2*A^6 + 30^2*A^7 + 15^2*A^8 + 5^2*A^9 + A^10)*x^5/5 +... which involves the squares of the trinomial coefficients A027907(n,k).
Programs
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PARI
{a(n)=local(A=1+x); A=1/x*serreverse(x*(1-x)*(1-x^3)*(1-x^4)/(1-x^12+x*O(x^n))); polcoeff(A, n)} -
PARI
/* G.f. A(x) using the squares of the trinomial coefficients */ {A027907(n, k)=polcoeff((1+x+x^2)^n, k)} {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(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 + x^4)/(1-x)^2.
G.f.: A(x) = (1/x)*Series_Reversion( x*(1-x)*(1-x^3)*(1-x^4)/(1-x^12) ).
Comments