A300041 - OEIS

%I #8 Feb 24 2018 12:50:35

%S 1,1,2,3,7,20,63,215,783,2998,11977,49656,212738,938836,4257792,

%T 19808597,94405713,460412410,2295740045,11695447378,60837384509,

%U 322968172763,1748975296265,9657311996480,54350006350292,311630231535041,1819713622889812,10817233370816701,65434087495967354,402615569685977397,2518832660928798529,16016013141937173805

%N G.f. satisfies: A(x) = Sum_{n>=0} x^n * (1 + x*A(x)^(n-1))^n.

%C Compare g.f. to a g.f. C(x) of the Catalan sequence:

%C C(x) = Sum_{n>=0} x^n*(1 + x*C(x)^2)^n where C(x) = 1 + x*C(x)^2.

%H Paul D. Hanna, <a href="/A300041/b300041.txt">Table of n, a(n) for n = 0..100</a>

%F G.f. satisfies:

%F (1) A(x) = Sum_{n>=0} x^n * (1 + x*A(x)^(n-1))^n.

%F (2) A(x) = Sum_{n>=0} x^(2*n) * A(x)^(n*(n-1)) / (1 - x*A(x)^n)^(n+1).

%F (3) A(x) = x/Series_Reversion( x*G(x) ), where G(x) = A(x*G(x)) is the g.f. of A300043.

%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 20*x^5 + 63*x^6 + 215*x^7 + 783*x^8 + 2998*x^9 + 11977*x^10 + 49656*x^11 + 212738*x^12 + ...

%e such that

%e A(x) = 1 + x*(1+x) + x^2*(1+x*A(x))^2 + x^3*(1+x*A(x)^2)^3 + x^4*(1+x*A(x)^3)^4 + x^5*(1+x*A(x)^4)^5 + x^6*(1+x*A(x)^5)^6 + ...

%e The g.f. also satisfies the series identity:

%e A(x) = 1/(1-x) + x^2/(1-x*A(x))^2 + x^4*A(x)^2/(1-x*A(x)^2)^3 + x^6*A(x)^6/(1-x*A(x)^3)^4 + x^8*A(x)^12/(1-x*A(x)^4)^5 + x^10*A(x)^20/(1-x*A(x)^5)^6 + ...

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*(1+x*(A+x*O(x^n))^(m-1))^m)); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(k=0, n, x^(2*k)*A^(k*(k-1))/(1 - x*A^k +x*O(x^n))^(k+1) )); polcoeff(A, n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A186998, A300042, A300043.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 24 2018