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%I #2 Mar 30 2012 18:37:22
%S 1,1,6,78,1648,49500,1957968,97097336,5834581632,414370221696,
%T 34127635732800,3211425586911168,341164552018811904,
%U 40517022329819203584,5335290940894955228160,773591071307555130451200
%N E.g.f. satisfies: A(x) = A(x*A(x))^2 - x*A'(x).
%F E.g.f. satisfies: x*A(x)^2 equals the g.f. of column 0 in the matrix log of the Riordan array (A(x), x*A(x)).
%F E.g.f.: A(x) = G(x)/x where G(x) = e.g.f. of A179497.
%F Let G_n(x) denote the n-th iteration of x*A(x) with G_0(x)=x, then
%F . [G_{n+1}(x)/x]^2 = A(x)^2*G_n'(x) for all n,
%F and L=x*A(x)^2 satisfies the series:
%F . A(x) = 1 + L + L*Dx(L)/2! + L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! +...
%F . G_{-1}(x)/x = 1 - L + L*Dx(L)/2! - L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! -+...
%F . G_n(x)/x = 1 + n*L + n^2*L*Dx(L)/2! + n^3*L*Dx(L*Dx(L))/3! + n^4*L*Dx(L*Dx(L*Dx(L)))/4! +...
%F where Dx(F) = d/dx(x*F).
%e E.g.f.: A(x) = 1 + x + 6*x^2/2! + 78*x^3/3! + 1648*x^4/4! + 49500*x^5/5! +...
%e Related expansions:
%e . x*A(x) = x + 2*x^2/2! + 18*x^3/3! + 312*x^4/4! + 8240*x^5/5! +...
%e . x*A(x)^2 = x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 20680*x^5/5! +..
%e . x*A'(x) = x + 12*x^2/2! + 234*x^3/3! + 6592*x^4/4! + 247500*x^5/5! +...
%e . A(x*A(x)) = 1 + x + 8*x^2/2! + 132*x^3/3! + 3400*x^4/4! + 120940*x^5/5! +...
%e . A(x*A(x))^2 = 1 + 2*x + 18*x^2/2! + 312*x^3/3! + 8240*x^4/4! + 297000*x^5/5! +...
%e Illustrate the iterations G_n(x) of G(x) = x*A(x) by:
%e . [G_3(x)/x]^2 = A(x)^2 * G_2'(x);
%e . [G_4(x)/x]^2 = A(x)^2 * G_3'(x);
%e . [G_5(x)/x]^2 = A(x)^2 * G_4'(x); ...
%e which can be shown by the chain rule of differentiation.
%e ...
%e The RIORDAN ARRAY (A(x), x*A(x)) begins:
%e . 1;
%e . 1, 1;
%e . 6/2!, 2, 1;
%e . 78/3!, 14/2!, 3, 1;
%e . 1648/4!, 192/3!, 24/2!, 4, 1;
%e . 49500/5!, 4136/4!, 348/3!, 36/2!, 5, 1;
%e . 1957968/6!, 124840/5!, 7680/4!, 552/3!, 50/2!, 6, 1;
%e . 97097336/7!, 4928256/6!, 233940/5!, 12520/4!, 810/3!, 66/2!, 7, 1; ...
%e where the g.f. of column k = A(x)^(k+1) for k>=0. ...
%e The MATRIX LOG of the above Riordan array (A(x), x*A(x)) begins:
%e . 0;
%e . 1, 0;
%e . 4/2!, 2, 0;
%e . 42/3!, 8/2!, 3, 0;
%e . 768/4!, 84/3!, 12/2!, 4, 0;
%e . 20680/5!, 1536/4!, 126/3!, 16/2!, 5, 0;
%e . 749040/6!, 41360/5!, 2304/4!, 168/3!, 20/2!, 6, 0;
%e . 34497792/7!, 1498080/6!, 62040/5!, 3072/4!, 210/3!, 24/2!, 7, 0; ...
%e where the g.f. of column k = (k+1)*x*A(x)^2 for k>=0.
%o (PARI) {a(n)=local(A=1+x+sum(m=2,n-1,a(m)*x^m/m!)+x*O(x^(n+5)));if(n<2,n!*polcoeff(A,n),n!*polcoeff(subst(A,x,x*A)^2,n)/(n-1))}
%Y Cf. A179497, A179499, A179421.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 31 2010
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