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%I #2 Mar 30 2012 18:37:22
%S 0,1,2,12,108,1420,24660,541968,14547792,465228720,17385553440,
%T 747776581200,36566808933600,2012537262763872,123612631608883872,
%U 8412289268206662720,630378349868153698560,51733701375836221013760
%N E.g.f. A(x) satisfies: L(x) = A(x)/(x*A'(x)) * L(A(x)) where L(x) = x + x*A(x).
%F E.g.f. satisfies: A(A(x)) = -1 + (1 + A(x))*A'(x)*x^2/A(x)^2.
%F Let A_n(x) denote the n-th iteration of e.g.f. A(x), then
%F . A_{n+1}(x) = -1 + (1 + A(x))*A_n'(x)*x^2/A_n(x)^2.
%F . L(x) = A_n(x)/(x*A_n'(x)) * L(A_n(x)) where L(x) = x + x*A(x).
%F ...
%F Let L = L(x) = x + x*A(x), then:
%F . A(x)/x = 1 + L + L*Dx(L)/2! + L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! +...
%F . A_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) = x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1420*x^5/5! +...
%e Related expansions:
%e . L(x) = x + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 540*x^5/5! +...
%e . L(A(x)) = x + 4*x^2/2! + 30*x^3/3! + 348*x^4/4! + 5560*x^5/5! +...
%e . x*A'(x) = x + 4*x^2/2! + 36*x^3/3! + 432*x^4/4! + 7100*x^5/5! +...
%e . A(x)/x = 1 + x + 4*x^2/2! + 27*x^3/3! + 284*x^4/4! + 4110*x^5/5! +...
%e where L(x) = x + x*A(x) = A(x)/(x*A'(x)) * L(A(x)).
%e ...
%e The RIORDAN ARRAY (A(x)/x, A(x)) begins:
%e 1;
%e 1, 1;
%e 4/2!, 2, 1;
%e 27/3!, 10/2!, 3, 1;
%e 284/4!, 78/3!, 18/2!, 4, 1;
%e 4110/5!, 880/4!, 159/3!, 28/2!, 5, 1;
%e 77424/6!, 13220/5!, 1932/4!, 276/3!, 40/2!, 6, 1;
%e 1818474/7!, 252828/6!, 30390/5!, 3608/4!, 435/3!, 54/2!, 7, 1; ...
%e where the g.f. of column k = A(x)^(k+1)/x^k for k>=0.
%e ...
%e The MATRIX LOG of the above Riordan array (A(x)/x, A(x)) begins:
%e 0;
%e 1, 0;
%e 2/2!, 2, 0;
%e 6/3!, 4/2!, 3, 0;
%e 48/4!, 12/3!, 6/2!, 4, 0;
%e 540/5!, 96/4!, 18/3!, 8/2!, 5, 0;
%e 8520/6!, 1080/5!, 144/4!, 24/3!, 10/2!, 6, 0;
%e 172620/7!, 17040/6!, 1620/5!, 192/4!, 30/3!, 12/2!, 7, 0; ...
%e where the g.f. of column k = (k+1)*(x + x*A(x)) for k>=0.
%e ...
%e To illustrate the inversion series, let L=L(x)=x + x*A(x), then:
%e . A(A(x)) = x + 4*x^2/2! + 36*x^3/3! + 480*x^4/4! + 8720*x^5/5! +...
%e . A(A(x))/x = 1 + 2*L + 2^2*L*Dx(L)/2! + 2^3*L*Dx(L*Dx(L))/3! +...
%e . A_3(x) = x + 6*x^2/2! + 72*x^3/3! + 1260*x^4/4! + 29340*x^5/5! +...
%e . A_3(x)/x = 1 + 3*L + 3^2*L*Dx(L)/2! + 3^3*L*Dx(L*Dx(L))/3! +...
%e where Dx(F) = d/dx(x*F).
%o (PARI) {a(n)=local(A=[1,1]);for(i=2,n, A=concat(A,0);G=x*Ser(A);A[ #A]=polcoeff(1+subst(G,x,G)+O(x^#A)-(1+G)*deriv(G)*x^2/G^2,#A-1)/(#A-2));if(n<1,0,n!*A[n])}
%Y Cf. A179494, A179420, A179421.
%K eigen,nonn
%O 0,3
%A _Paul D. Hanna_, Jul 23 2010
%E Typos in formula and example corrected by _Paul D. Hanna_, Jul 28 2010
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