A177383 - OEIS

%I #12 Mar 06 2020 17:14:21

%S 1,1,-3,18,-145,1398,-15330,186004,-2455893,34903110,-529903726,

%T 8547005820,-145847288730,2624318836412,-49658727391620,

%U 985921121129832,-20496980960236365,445413655950148710,-10100793273684125430

%N G.f. satisfies: [x^n] A(x)^(n+1) = [x^n] A(x)^(n+2) for n>1 with A'(0)=A(0)=1.

%H Vaclav Kotesovec, <a href="/A177383/b177383.txt">Table of n, a(n) for n = 0..400</a>

%F G.f. satisfies: A(x) = 1 + x * (A(x) - x*A'(x)) / A(x)^3. - _Paul D. Hanna_, Aug 04 2014

%F Let G(x) be the g.f. of A177384, then

%F . a(n) = [x^n] G(x)^(-n+1)/(-n+1) for n>1,

%F . A(x) = G(x/A(x)) so that A(x) = x/Series_Reversion(x*G(x))

%F where G(x) = 1 + x/(G(x) + x*G'(x)).

%F a(n) ~ c * (-1)^(n+1) * n! * n^4, where c = 0.0147556981601927885048672335828437... (same as for A182304). - _Vaclav Kotesovec_, Jul 28 2014

%F a(n) / A177384(n) ~ exp(1). - _Vaclav Kotesovec_, Mar 06 2020

%e G.f.: A(x) = 1 + x - 3*x^2 + 18*x^3 - 145*x^4 + 1398*x^5 +...

%e Coefficients in the initial powers of A(x) begin:

%e [1,. 1, -3,. 18,. -145,. 1398,. -15330,. 186004,. -2455893,...];

%e [1,. 2, -5,. 30,. -245,. 2398,. -26670,. 327740,. -4376445,...];

%e [1,. 3,(-6), 37,. -309,. 3081,. -34806,. 433470,. -5855301,...];

%e [1,. 4,(-6),(40), -345,. 3516,. -40398,. 510120,. -6971325,...];

%e [1,. 5, -5, (40),(-360), 3761,. -43995,. 563460,. -7790675,...];

%e [1,. 6, -3,. 38, (-360),(3864), -46049,. 598266,. -8368635,...];

%e [1,. 7,. 0,. 35,. -350, (3864),(-46928), 618465,. -8751225,...];

%e [1,. 8,. 4,. 32,. -334,. 3792, (-46928),(627264), -8976609,...];

%e [1,. 9,. 9,. 30,. -315,. 3672,. -46284, (627264),(-9076320),...];

%e [1, 10, 15,. 30,. -295,. 3522,. -45180,. 620560, (-9076320),...];

%e where the above terms in parenthesis illustrate the property

%e that the coefficients of x^n in A(x)^(n+1) and A(x)^(n+2) are equal.

%e Also, the same terms derive A177384(n) = [x^n] A(x)^(n+1)/(n+1):

%e [1, 1, -2, 10, -72, 644, -6704, 78408, -1008480, 14065744, ...]

%e where g.f. G(x) of A177384 satisfies 1/x*d/dx[xG(x)]^2/2 = x+d/dx xG(x):

%e d/dx x*G(x) = 1 + 2*x - 6*x^2 + 40*x^3 - 360*x^4 + 3864*x^5 -...

%e d/dx [x*G(x)]^2/2 = x + 3*x^2 - 6*x^3 + 40*x^4 - 360*x^5 + 3864*x^6 -...

%o (PARI) {a(n)=local(G=1+x+x*O(x^n),H);for(i=1,n,G=1+x/(G+x*deriv(G)+x*O(x^n))); H=x/serreverse(x*G);polcoeff(H,n)}

%o for(n=0,20,print1(a(n),", "))

%o (PARI) /* From A(x) = 1 + x*(A(x) - x*A'(x)) / A(x)^3 : */

%o {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*(A-x*A')/(A^3 +x*O(x^n)));polcoeff(A,n)}

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A177384, A158883, A182304.

%K sign

%O 0,3

%A _Paul D. Hanna_, May 15 2010