A325213 G.f.: A(x) = exp( Sum_{n>=1} (x^n/n) * (1 + x^n*A(x)^n) / (1 - x^n*A(x)^n)).

1, 1, 3, 7, 22, 68, 230, 790, 2809, 10151, 37346, 139116, 524160, 1993184, 7641077, 29497043, 114568292, 447397474, 1755543081, 6918285937, 27369693199, 108659122477, 432762724820, 1728625913332, 6923326780630, 27797059897118, 111859099064653, 451086075723925, 1822628515630354, 7377839562543650

Offset: 0

Author

Paul D. Hanna, Apr 16 2019

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 22*x^4 + 68*x^5 + 230*x^6 + 790*x^7 + 2809*x^8 + 10151*x^9 + 37346*x^10 + 139116*x^11 + 524160*x^12 + ...
satisfies
log(A(x)) = x*(1+x*A(x))/(1-x*A(x)) + x^2/2*(1+x^2*A(x)^2)/(1-x^2*A(x)^2) + x^3/3*(1+x^3*A(x)^3)/(1-x^3*A(x)^3) + x^4/4*(1+x^4*A(x)^4)/(1-x^4*A(x)^4) + x^5/5*(1+x^5*A(x)^5)/(1-x^5*A(x)^5) + x^6/6*(1+x^6*A(x)^6)/(1-x^6*A(x)^6) + ...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 13*x^3/3 + 53*x^4/4 + 191*x^5/5 + 761*x^6/6 + 2969*x^7/7 + 11893*x^8/8 + 47677*x^9/9 + 192935*x^10/10 + 783575*x^11/11 + 3197657*x^12/12 + ...
		

Crossrefs

Cf. A210043.

Programs

PARI

{a(n) = my(A=[1]); for(i=1, n, A=Vec( exp( sum(m=1, #A+1, x^m/m * (1 + x^m*Ser(A)^m)/(1 - x^m*Ser(A)^m) ) )) ); A[n+1]}
for(n=0, 40, print1(a(n), ", "))