A295762 - OEIS

%I #16 Nov 03 2019 04:54:39

%S 1,3,12,69,444,3060,22104,165195,1266636,9908196,78760920,634379124,

%T 5166150000,42465716328,351876854448,2936058188877,24648274487556,

%U 208040487845076,1764376309044792,15027939263874132,128495423551583664,1102547377746843624,9490542912076091184,81931260285359287812,709199467337528862768,6153967855892699398368,53521531522907694320928,466461452477641527148344

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

%C Odd terms seem to occur only at a(2^n) for n>=0 (conjecture).

%H Paul D. Hanna, <a href="/A295762/b295762.txt">Table of n, a(n) for n = 1..1030</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = x + 3 * A( (x + 2*A(x))^2/9 ).

%F (2) A(x) = -x/2 + 3/2*Series_Reversion( x - 2*A(x^2) ).

%F (3) x = A( -2*x + 3 * Series_Reversion( x + A(x^2) ) ).

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

%F a(n) ~ c * d^n / n^(3/2), where d = 9.2093789571628170578048631587867514135871993630966974111785256401203919511... and c = 0.0682994378249920600052979267397719638823914060552824374... - _Vaclav Kotesovec_, Dec 04 2017

%e G.f.: A(x) = x + 3*x^2 + 12*x^3 + 69*x^4 + 444*x^5 + 3060*x^6 + 22104*x^7 + 165195*x^8 + 1266636*x^9 + 9908196*x^10 + 78760920*x^11 + 634379124*x^12 +...

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

%e RELATED SERIES.

%e A(x - 2*A(x^2)) = x + x^2 + 3*x^4 + 12*x^6 + 69*x^8 + 444*x^10 + 3060*x^12 + 22104*x^14 + 165195*x^16 + 1266636*x^18 + 9908196*x^20 +...

%e which equals x + A(x^2).

%e Series_Reversion( x - 2*A(x^2) ) = x + 2*x^2 + 8*x^3 + 46*x^4 + 296*x^5 + 2040*x^6 + 14736*x^7 + 110130*x^8 + 844424*x^9 + 6605464*x^10 + 52507280*x^11 + 422919416*x^12 +...

%e which equals (2*A(x) + x)/3.

%e A( (x + 2*A(x))^2/9 ) = x^2 + 4*x^3 + 23*x^4 + 148*x^5 + 1020*x^6 + 7368*x^7 + 55065*x^8 + 422212*x^9 + 3302732*x^10 + 26253640*x^11 + 211459708*x^12 +...

%e which equals (A(x) - x)/3.

%e Odd terms seem to occur only at positions 2^n, n>=0, beginning:

%e [1, 3, 69, 165195, 2936058188877, 2740954751925406954539018771, 6899036855844990995854505818787102393537422152828959745477, ...].

%t nmax = 28; sol = {a[1] -> 1}; Do[A[x_] = Sum[a[k] x^k, {k, 1, n}] /. sol; eq = CoefficientList[A[x - 2 A[x^2]] - (x + A[x^2]) + O[x]^(n+1) // Normal, x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];

%t sol /. Rule -> Set;

%t a /@ Range[nmax] (* _Jean-François Alcover_, Nov 03 2019 *)

%o (PARI) {a(n) = my(A=x); for(i=1,n, A = -x/2 + 3/2*serreverse(x - 2*subst(A,x,x^2) +x^2*O(x^n)) ); polcoeff(A,n)}

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

%Y Cf. A141200, A295760, A295761.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Dec 03 2017