A274479 - OEIS

%I #15 Aug 01 2016 18:00:22

%S 1,1,4,10,34,106,361,1219,4252,14932,53263,191533,695233,2540617,

%T 9344050,34546672,128330533,478653973,1791816967,6729202603,

%U 25344884479,95707901503,362269464487,1374203633335,5223097370170,19888174932226,75856437036451,289780169876749,1108607284380835,4246966803249139,16290547536335716,62562701811659506,240540845892246253,925825162823212429,3567069859670052457,13756707569545384033

%N G.f. satisfies: A(x)^2 = A( x^2/(1 - 2*x - 4*x^2) ).

%C Compare g.f. with the identities:

%C (1) F(x)^2 = F( x^2/(1 - 2*x + 2*x^2) ) when F(x) = x/(1-x).

%C (2) M(x)^2 = M( x^2/(1 - 2*x) ) when M(x) = (1-x - sqrt(1-2*x-3*x^2))/(2*x) is a g.f. of the Motzkin numbers (A001006).

%C a(n) = 1 (mod 3) for n>=1 (conjecture).

%C Radius of convergence of g.f. A(x) is r = 1/4 where r = r^2/(1-2*r-4*r^2) with A(1/4) = 1.

%C What is the limit a(n)/A000108(n) ?  Note that A000108(n) = binomial(2*n,n)/(n+1) is the n-th Catalan number.

%H Paul D. Hanna, <a href="/A274479/b274479.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) satisfies: A( x/(1 + x + 3*x^2) )^2 = A( x^2/(1 + x^2 + 9*x^4) ).

%F Let G(x) denote the g.f. of A264412, where G(x)^2 = G(x^2) + 6*x, then g.f. A(x) satisfies:

%F (1) A(x) = x/(1-x) * G( A(x)^2 ),

%F (2) G(x^2) = x/Series_Reversion(A(x)) - x,

%F (3) A( x/(G(x^2) + x) ) = x,

%F (4) A(x)^2/(G(A(x)^4) + A(x)^2) = x^2/(1 - 2*x - 4*x^2).

%e G.f.: A(x) = x + x^2 + 4*x^3 + 10*x^4 + 34*x^5 + 106*x^6 + 361*x^7 + 1219*x^8 + 4252*x^9 + 14932*x^10 + 53263*x^11 + 191533*x^12 +...

%e such that A( x^2/(1-2*x-4*x^2) ) = A(x)^2.

%e RELATED SERIES.

%e A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 28*x^5 + 104*x^6 + 360*x^7 + 1306*x^8 + 4688*x^9 + 17106*x^10 + 62548*x^11 + 230570*x^12 + 853512*x^13 + 3176161*x^14 + 11866142*x^15 +...

%e The series reversion of the g.f. A(x) begins:

%e Series_Reversion(A(x)) = x - x^2 - 2*x^3 + 5*x^4 + 4*x^5 - 22*x^6 - 5*x^7 + 95*x^8 - 17*x^9 - 412*x^10 + 220*x^11 + 1790*x^12 - 1559*x^13 - 7771*x^14 +...

%e which is related to A264412 by:

%e x/Series_Reversion(A(x)) = 1 + x + 3*x^2 - 3*x^4 + 9*x^6 - 33*x^8 + 126*x^10 - 513*x^12 + 2214*x^14 - 9876*x^16 + 45045*x^18 - 209493*x^20 +...+ A264412(n)*x^(2*n) +...

%e The g.f. G(x) of A264412 begins:

%e G(x) = 1 + 3*x - 3*x^2 + 9*x^3 - 33*x^4 + 126*x^5 - 513*x^6 + 2214*x^7 - 9876*x^8 + 45045*x^9 - 209493*x^10 +...

%e where G(x)^2 = G(x^2) + 6*x.

%e Also, we have A(x/(1 + x + 3*x^2))^2 = A(x^2/(1 + x^2 + 9*x^4)), where the series begin:

%e A(x/(1 + x + 3*x^2)) = x - 3*x^5 + 3*x^9 + 81*x^13 - 840*x^17 + 3960*x^21 + 711*x^25 - 152145*x^29 + 1009254*x^33 - 1772820*x^37 + 1991277*x^41 +...

%e A(x^2/(1 + x^2 + 9*x^4)) = x^2 - 6*x^6 + 15*x^10 + 144*x^14 - 2157*x^18 + 13446*x^22 - 20817*x^26 - 420876*x^30 + 4282764*x^34 - 17051652*x^38 +...

%e which is equal to A(x/(1 + x + 3*x^2))^2.

%o (PARI) {a(n) = my(A=x); for(i=1, #binary(n+1), A = sqrt( subst(A, x, x^2/(1-2*x-4*x^2 +x*O(x^n)) ) ) ); polcoeff(A, n)}

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A264412, A274478, A274484.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jul 27 2016