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A299042 G.f.: Sum_{n>=0} x^(n^2) * C(x^n)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

%I A299042 #16 Feb 17 2018 12:38:37
%S A299042 1,1,1,2,6,14,44,132,434,1431,4876,16796,58831,208012,743032,2674449,
%T A299042 9695275,35357670,129646248,477638700,1767268056,6564120510,
%U A299042 24466283816,91482563640,343059672747,1289904147325,4861946609464,18367353073153,69533551658952,263747951750360,1002242219329245,3814986502092304,14544636048921919,55534064877060132,212336130447600780
%N A299042 G.f.: Sum_{n>=0} x^(n^2) * C(x^n)^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%C A299042 Compare to: Sum{n>=0} Series_Reversion( x/(1 + x^n)^(1/n) )^(n^2)  =  Sum_{n>=0} x^(n^2)/(1 - x^n)^n, the g.f. of A143862.
%H A299042 Paul D. Hanna, <a href="/A299042/b299042.txt">Table of n, a(n) for n = 0..400</a>
%F A299042 G.f.: Sum{n>=0} (1 - sqrt(1 - 4*x^n))^n / 2^n.
%F A299042 G.f.: Sum{n>=0} Series_Reversion( x*(1 - x^n)^(1/n) )^(n^2).
%e A299042 G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 14*x^5 + 44*x^6 + 132*x^7 + 434*x^8 + 1431*x^9 + 4876*x^10 + 16796*x^11 + 58831*x^12 + ...
%e A299042 such that
%e A299042 A(x) = 1 + (1 - sqrt(1 - 4*x))/2 + (1 - sqrt(1 - 4*x^2))^2/2^2 + (1 - sqrt(1 - 4*x^3))^3/2^3 + (1 - sqrt(1 - 4*x^4))^4/2^4 + (1 - sqrt(1 - 4*x^5))^5/2^5 + (1 - sqrt(1 - 4*x^6))^6/2^6 + ...
%e A299042 The related series x^(n^2) * C(x^n)^n = (1 - sqrt(1 - 4*x^n))^n/2^n begin:
%e A299042 n=1: x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + ...;
%e A299042 n=2: x^4 + 2*x^6 + 5*x^8 + 14*x^10 + 42*x^12 + ...;
%e A299042 n=3: x^9 + 3*x^12 + 9*x^15 + 28*x^18 + 90*x^21 + ...;
%e A299042 n=4: x^16 + 4*x^20 + 14*x^24 + 48*x^28 + 165*x^32 + ...;
%e A299042 n=5: x^25 + 5*x^30 + 20*x^35 + 75*x^40 + 275*x^45 + ...;
%e A299042 n=6: x^36 + 6*x^42 + 27*x^48 + 110*x^54 + 429*x^60 + ...;
%e A299042 ...
%e A299042 SPECIFIC VALUES.
%e A299042 A(1/4) = Sum_{n>=0} (2^(n-1) - sqrt(4^(n-1) - 1))^n / 2^(n^2) = 1.504491300666... = 1 + 1/2 + (2 - sqrt(3))^2/2^4 + (4 - sqrt(15))^3/2^9 + (8 - sqrt(63))^4/2^16 + (16 - sqrt(255))^5/2^25 + (32 - sqrt(1023))^6/2^36 + (64 - sqrt(4095))^7/2^49 + ...
%e A299042 A(-1/4) = Sum_{n>=0} (2^(n-1) - sqrt(4^(n-1) + 1))^n / 2^(n^2) = 0.79637258079... = 1 + (1 - sqrt(2))/2 + (2 - sqrt(5))^2/2^4 + (4 - sqrt(17))^3/2^9 + (8 - sqrt(65))^4/2^16 + (16 - sqrt(257))^5/2^25 + (32 - sqrt(1025))^6/2^36 + ...
%o A299042 (PARI) {a(n) = my(A); A = sum(m=0,sqrtint(n+1), (1 - sqrt(1 - 4*x^m +x*O(x^n) ))^m / 2^m); polcoeff(A,n)}
%o A299042 for(n=0,40,print1(a(n),", "))
%Y A299042 Cf. A000108.
%K A299042 nonn
%O A299042 0,4
%A A299042 _Paul D. Hanna_, Feb 16 2018