A292814 - OEIS

%I #15 Oct 09 2017 09:07:39

%S 1,4,-40,1400,-68000,4350000,-335400000,30085500000,-3064470000000,

%T 348742875000000,-43821408750000000,6025223212500000000,

%U -899952237525000000000,145148307600812500000000,-25148944415520000000000000,4660143903279037500000000000,-919883216910828187500000000000,192745988160330338593750000000000,-42734275076178265882812500000000000,9996684603960098310695312500000000000,-2460868592009779893077500000000000000000

%N G.f.: A(x) satisfies: A( 6*x - 5*A(x) ) = x - 16*x^2.

%C Conjecture: In general, if k > 0 and g.f. A(x) satisfies A((k+2)*x - (k+1)*A(x)) = x - (k*x)^2, then a(n) ~ (-1)^n * c(k) * (k*(k+1)/log(k+1))^n * n! / n^((k-1)/(k+1) + (k-2)*log(k+1)/k), where c(k) is a constant. - _Vaclav Kotesovec_, Oct 09 2017

%H Vaclav Kotesovec, <a href="/A292814/b292814.txt">Table of n, a(n) for n = 1..300</a> (terms 1..200 from Paul D. Hanna)

%F a(n) ~ (-1)^n * c * 20^n * n! / (n^(3/5 + log(5)/2) * (log(5))^n), where c = 0.04310841216382017737... - _Vaclav Kotesovec_, Oct 09 2017

%e G.f.: A(x) = x + 4*x^2 - 40*x^3 + 1400*x^4 - 68000*x^5 + 4350000*x^6 - 335400000*x^7 + 30085500000*x^8 - 3064470000000*x^9 + 348742875000000*x^10 - 43821408750000000*x^11 + 6025223212500000000*x^12 - 899952237525000000000*x^13 + 145148307600812500000000*x^14 - 25148944415520000000000000*x^15 +...

%e such that A( 6*x - 5*A(x) ) = x - 16*x^2.

%e RELATED SERIES.

%e Define Ai(x) such that Ai(A(x)) = x, then Ai(x) begins:

%e Ai(x) = x - 4*x^2 + 72*x^3 - 2520*x^4 + 123424*x^5 - 7710448*x^6 + 579543872*x^7 - 50689868256*x^8 + 5046335530880*x^9 - 562904575158208*x^10 +...

%e where Ai(x - 16*x^2) = 6*x - 5*A(x).

%o (PARI) {a(n) = my(A=x,V=[1,4]); for(i=1,n, V = concat(V,0); A=x*Ser(V); V[#V] = Vec( subst(A,x, 6*x - 5*A) )[#V]/4 );V[n]}

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

%Y Cf. A291198, A292812, A292813, A292811, A293454, A293455, A293456.

%K sign

%O 1,2

%A _Paul D. Hanna_, Sep 23 2017