A317997 - OEIS

%I #17 Oct 08 2019 22:39:55

%S 1,1,3,8,22,62,184,566,1779,5678,18367,60091,198454,660659,2214700,

%T 7469730,25330135,86308528,295349232,1014609122,3497707471,

%U 12096300263,41955256710,145908382056,508676792788,1777418947825,6223737747078,21835417474588,76747151840311,270209860057674,952864014673991,3365181832760454,11901315653535352

%N G.f. A(x) satisfies: Sum_{n>=0} ( x^n + (-1)^n*A(x) )^n = 1.

%H Paul D. Hanna, <a href="/A317997/b317997.txt">Table of n, a(n) for n = 1..520</a>

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

%F (1) 1 = Sum_{n>=0} ( x^n + (-1)^n*A(x) )^n.

%F (2) 1 = Sum_{n>=0} x^(n^2) / (1 + (-x)^n*A(x))^(n+1).

%F a(n) ~ c * d^n / n^(3/2), where d = 3.699932925974338637547631722109256130661... and c = 0.41429291356124695745726003609883876... - _Vaclav Kotesovec_, Sep 06 2018

%F A(1/d) = 0.8093560777550741... where d is given above. - _Paul D. Hanna_, Oct 08 2019

%e G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 22*x^5 + 62*x^6 + 184*x^7 + 566*x^8 + 1779*x^9 + 5678*x^10 + 18367*x^11 + 60091*x^12 + ...

%e such that

%e 1 = 1 + (x - A(x)) + (x^2 + A(x))^2 + (x^3 - A(x))^3 + (x^4 + A(x))^4 + (x^5 - A(x))^5 + (x^6 + A(x))^6 + (x^7 - A(x))^7 + ...

%e Also

%e 1 = 1/(1 + A(x)) + x/(1 - x*A(x))^2 + x^4/(1 + x^2*A(x))^3 + x^9/(1 - x^3*A(x))^4 + x^16/(1 + x^4*A(x))^5 + x^25/(1 - x^5*A(x))^6 + x^36/(1 + x^6*A(x))^7 + ...

%o (PARI) /* Using: 1 = Sum_{n>=0} ( x^n + (-1)^n*A(x) )^n */

%o {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); A[#A] = polcoeff( sum(m=1,#A, (x^m + (-1)^m*x*Ser(A))^m ), #A); ); polcoeff(A,n)}

%o for(n=1,35,print1(a(n),", "))

%o (PARI) /* Faster: 1 = Sum_{n>=0} x^(n^2) / (1 + (-x)^n*A(x))^(n+1) */

%o {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = polcoeff( sum(m=0, sqrtint(#A+1), x^(m^2)/(1 + (-x)^m*x*Ser(A))^(m+1) ), #A); ); polcoeff(A, n)}

%o for(n=1, 35, print1(a(n), ", "))

%Y Cf. A247332.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Aug 31 2018