A274736 - OEIS

%I #7 Jul 19 2016 22:51:03

%S 1,2,11,68,497,3862,31691,269064,2346455,20883170,188928097,

%T 1732200684,16059878311,150308726166,1418213573009,13475591449360,

%U 128831965006038,1238387413610926,11961509281268996,116035914722431492,1130027817795205818,11043733824515222466,108276550546410148128,1064692398520050464088,10497351182518154842770,103754825395109251885646,1027845237135955778148684

%N G.f. A(x) satisfies: A(x)^2 - 4*A(x)^3 = A(x^2).

%H Paul D. Hanna, <a href="/A274736/b274736.txt">Table of n, a(n) for n = 1..300</a>

%F G.f. A(x) satisfies: A(B(x)^2) = x^2 - 4*x^3, where A(B(x)) = x.

%e G.f.: A(x) = x + 2*x^2 + 11*x^3 + 68*x^4 + 497*x^5 + 3862*x^6 + 31691*x^7 + 269064*x^8 + 2346455*x^9 + 20883170*x^10 +...

%e such that A(x)^2 - 4*A(x)^3 = A(x^2).

%e RELATED SERIES.

%e A(x)^2 = x^2 + 4*x^3 + 26*x^4 + 180*x^5 + 1387*x^6 + 11208*x^7 + 94388*x^8 + 817448*x^9 + 7238609*x^10 + 65220372*x^11 + 596019486*x^12 +...

%e A(x)^3 = x^3 + 6*x^4 + 45*x^5 + 344*x^6 + 2802*x^7 + 23580*x^8 + 204362*x^9 + 1809528*x^10 + 16305093*x^11 + 149003906*x^12 +...

%e The square root of A(x^2) is an integer series:

%e sqrt(A(x^2)) = A(x)*sqrt(1 - 4*A(x)) = x + x^3 + 5*x^5 + 29*x^7 + 207*x^9 + 1579*x^11 + 12811*x^13 + 107823*x^15 + 934134*x^17 + 8269964*x^19 +...

%e Let B(x) denote the series reversion of g.f. A(x), so that A(B(x)) = x, where

%e B(x) = x - 2*x^2 - 3*x^3 + 2*x^4 - 18*x^5 - 12*x^6 - 151*x^7 - 222*x^8 - 948*x^9 - 2552*x^10 - 11484*x^11 - 28632*x^12 - 130776*x^13 - 396304*x^14 +...

%e Note that g.f. A(x) and B(x) satisfy:

%e (1) A(B(x)^2) = C(x) = x^2 - 4*x^3.

%e (2) A(B(x)^4) = C(C(x)) = x^4 - 8*x^5 + 12*x^6 + 48*x^7 - 192*x^8 + 256*x^9.

%e (3) A(B(x)^8) = C(C(C(x))).

%o (PARI) /* From A(B(x)^2) = x^2 - 4*x^3, where A(B(x)) = x: */

%o {a(n) = my(A=[1, 2], F, B); for(i=1, n, A=concat(A, 0); F=x*Ser(A); B=serreverse(F); A[#A] = Vec(subst(F, x, B^2))[#A]/2); A[n]}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A273095, A274737.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 19 2016