|
%I #8 Oct 22 2020 02:35:37
%S 1,1,2,13,176,3886,125374,5550713,323184220,23969935266,2208185433708,
%T 247518073493022,33184572994243884,5244964339235985636,
%U 965302981114301621022,204679102209593395617929
%N G.f. satisfies: [x^(n+1)] A(x)^n = n*(n+1)*{[x^n] A(x)^n} for n>=0.
%H Vaclav Kotesovec, <a href="/A183606/b183606.txt">Table of n, a(n) for n = 0..250</a>
%F G.f.: A(x) = x/Series_Reversion(G(x)) where A(x*G(x)) = G(x) satisfies: G(x) = 1/(1-x - x*{d/dx x^2*G'(x)/G(x)}) and is the g.f. of A183607.
%F a(n) ~ c * n!^2 / n, where c = 2.05242598709856847249724356196772... - _Vaclav Kotesovec_, Oct 22 2020
%e G.f.: A(x) = 1 + x + 2*x^2 + 13*x^3 + 176*x^4 + 3886*x^5 +...
%e A(x) satisfies A(x*G(x)) = G(x) where:
%e G(x) = 1 + x + 3*x^2 + 20*x^3 + 249*x^4 + 5087*x^5 + 155180*x^6 +...
%e which satisfies: G(x) = 1/(1-x - x*{d/dx x^2*G'(x)/G(x)}).
%e ...
%e The table of coefficients in the initial powers of g.f. A(x) begins:
%e A^0: [1,_0, 0, 0, 0, 0, 0, 0, ...];
%e A^1: [1, 1,_2, 13, 176, 3886, 125374, 5550713, 323184220, ...];
%e A^2: [1, 2, 5,_30, 382, 8176, 259393, 11372294, 658103374, ...];
%e A^3: [1, 3, 9, 52,_624, 12921, 402749, 17479176, 1005197070, ...];
%e A^4: [1, 4, 14, 80, 909,_18180, 556210, 23886840, 1364926338, ...];
%e A^5: [1, 5, 20, 115, 1245, 24021,_720630, 30611920, 1737775110, ...];
%e A^6: [1, 6, 27, 158, 1641, 30522, 896960,_37672320, 2124251838, ...];
%e A^7: [1, 7, 35, 210, 2107, 37772, 1086260, 45087344,_2524891264, ...];
%e ...
%e where the upper diagonal:
%e [0, 2, 30, 624, 18180, 720630, 37672320, 2524891264, ...]
%e is term-wise related to the main diagonal in the following way:
%e [0*1, 2*1, 6*5, 12*52, 20*909, 30*24021, 42*896960, 56*45087344, ...].
%o (PARI) {a(n)=local(A=[1,1]);for(i=1,n,A=concat(A,0);A[#A]=((#A-1)*(#A-2)*Vec(Ser(A)^(#A-2))[#A-1] - Vec(Ser(A)^(#A-2))[#A])/(#A-2));A[n+1]}
%Y Cf. A183607.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 13 2011
|
