A370538 - OEIS

%I #7 Mar 08 2024 16:06:38

%S 1,1,1,-3,15,3,148,-314,466,-1980,13410,-12348,52579,-312347,898033,

%T -221141,10087231,-27550253,35879866,-120326612,1290473218,

%U -1015145068,2822763419,-31671326569,92976002320,40883035636,903217721062,-3628100989426,1532499458666,-6655463860036

%N Expansion of g.f. A(x) = G( x*(1 + 2*x)*G(x) )^(1/2) = G( x*(1 + 3*x)*G(x)^2 )^(1/3), where G(x) is the g.f. of A370537.

%H Paul D. Hanna, <a href="/A370538/b370538.txt">Table of n, a(n) for n = 1..501</a>

%e G.f.: A(x) = x + x^2 + x^3 - 3*x^4 + 15*x^5 + 3*x^6 + 148*x^7 - 314*x^8 + 466*x^9 - 1980*x^10 + 13410*x^11 - 12348*x^12 + 52579*x^13 - 312347*x^14 + 898033*x^15 + ...

%e where A(x) = G( x*(1 + 2*x)*G(x) )^(1/2) = G( x*(1 + 3*x)*G(x)^2 )^(1/3),

%e and G(x) is the g.f. of  A370537, which begins

%e G(x) = x + 3*x^3 - 10*x^4 + 42*x^5 - 72*x^6 + 432*x^7 - 1296*x^8 + 3474*x^9 - 11644*x^10 + 48438*x^11 - 119532*x^12 + 385150*x^13 + ...

%e RELATED SERIES.

%e A(x)^2 = G( x*(1 + 2*x)*G(x) ) = x^2 + 2*x^3 + 3*x^4 - 4*x^5 + 25*x^6 + 30*x^7 + 341*x^8 - 416*x^9 + 807*x^10 - 4454*x^11 + 30125*x^12 + ...

%e A(x)^3 = G( x*(1 + 3*x)*G(x)^2 ) = x^3 + 3*x^4 + 6*x^5 - 2*x^6 + 33*x^7 + 75*x^8 + 607*x^9 - 189*x^10 + 1287*x^11 - 7143*x^12 + 48735*x^13 + ...

%e A(x)^6 = x^6 + 6*x^7 + 21*x^8 + 32*x^9 + 90*x^10 + 324*x^11 + 2064*x^12 + 4032*x^13 + 9513*x^14 - 6310*x^15 + 116499*x^16 + ...

%e where A(x)^6 = G( x*(1 + 2*x)*G(x) )^3 = G( x*(1 + 3*x)*G(x)^2 )^2.

%o (PARI) {a(n) = my(A,G, V=[1]); for(i=1, n+1, V = concat(V, 0); G = x*Ser(V);

%o V[#V] = polcoeff( subst(G, x, x*(1 + 2*x)*G )^3 - subst(G, x, x*(1 + 3*x)*G^2 )^2, #V+5); ); A = subst(G, x, x*(1 + 2*x)*G )^(1/2); polcoeff(A,n)}

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

%Y Cf. A370537, A370438.

%K sign

%O 1,4

%A _Paul D. Hanna_, Mar 08 2024