This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A179497 #4 Mar 30 2012 18:37:22
%S A179497 1,2,18,312,8240,297000,13705776,776778688,52511234688,4143702216960,
%T A179497 375403993060800,38537107042934016,4435139176244554752,
%U A179497 567238312617468850176,80029364113424328422400
%N A179497 E.g.f. satisfies: A(A(x))^2 = A(x)^2 * A'(x).
%H A179497 Paul D. Hanna, <a href="/A179497/b179497.txt">Table of n, a(n), n=1..100.</a>
%F A179497 E.g.f. A(x) satisfies: A(x)^2/x equals the e.g.f. of column 0 in the matrix log of the Riordan array (A(x)/x, A(x)).
%F A179497 Let A_n(x) denote the n-th iteration of e.g.f. A(x) with A_0(x)=x,
%F A179497 then L=A(x)^2/x satisfies the series:
%F A179497 . A(x)/x = 1 + L + L*Dx(L)/2! + L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! +...
%F A179497 . A_{-1}(x)/x = 1 - L + L*Dx(L)/2! - L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! -+...
%F A179497 . 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 A179497 where Dx(F) = d/dx(x*F).
%F A179497 Further, we have:
%F A179497 . [A_{n+1}(x)]^2 = A(x)^2*A_n'(x)
%F A179497 which holds for all n.
%e A179497 E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 312*x^4/4! + 8240*x^5/5! +..
%e A179497 Related expansions:
%e A179497 . A(x)/x = 1 + x + 6*x^2/2! + 78*x^3/3! + 1648*x^4/4! + 49500*x^5/5! +..
%e A179497 . A(x)^2/x = x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 20680*x^5/5! +..
%e A179497 . A'(x) = 1 + 2*x + 18*x^2/2! + 312*x^3/3! + 8240*x^4/4! +...
%e A179497 . A(A(x)) = x + 4*x^2/2! + 48*x^3/3! + 1008*x^4/4! + 30880*x^5/5! +...
%e A179497 . A(A(x))^2 = 2*x^2/2! + 24*x^3/3! + 480*x^4/4! + 13920*x^5/5! +...
%e A179497 Illustrate a main property of the iterations A_n(x) of A(x) by:
%e A179497 . [A_3(x)]^2 = A(x)^2 * A_2'(x);
%e A179497 . [A_4(x)]^2 = A(x)^2 * A_3'(x);
%e A179497 . [A_5(x)]^2 = A(x)^2 * A_4'(x); ...
%e A179497 which can be shown to hold by the chain rule of differentiation.
%e A179497 ...
%e A179497 The RIORDAN ARRAY (A(x)/x, A(x)) begins:
%e A179497 . 1;
%e A179497 . 1, 1;
%e A179497 . 6/2!, 2, 1;
%e A179497 . 78/3!, 14/2!, 3, 1;
%e A179497 . 1648/4!, 192/3!, 24/2!, 4, 1;
%e A179497 . 49500/5!, 4136/4!, 348/3!, 36/2!, 5, 1;
%e A179497 . 1957968/6!, 124840/5!, 7680/4!, 552/3!, 50/2!, 6, 1;
%e A179497 . 97097336/7!, 4928256/6!, 233940/5!, 12520/4!, 810/3!, 66/2!, 7, 1; ...
%e A179497 where the e.g.f. of column k = [A(x)/x]^(k+1) for k>=0.
%e A179497 ...
%e A179497 The MATRIX LOG of the above Riordan array (A(x)/x, A(x)) begins:
%e A179497 . 0;
%e A179497 . 1, 0;
%e A179497 . 4/2!, 2, 0;
%e A179497 . 42/3!, 8/2!, 3, 0;
%e A179497 . 768/4!, 84/3!, 12/2!, 4, 0;
%e A179497 . 20680/5!, 1536/4!, 126/3!, 16/2!, 5, 0;
%e A179497 . 749040/6!, 41360/5!, 2304/4!, 168/3!, 20/2!, 6, 0;
%e A179497 . 34497792/7!, 1498080/6!, 62040/5!, 3072/4!, 210/3!, 24/2!, 7, 0; ...
%e A179497 where the e.g.f. of column k = (k+1)*A(x)^2/x for k>=0.
%o A179497 (PARI) {a(n)=local(A=x+1*x^2+sum(m=3,n-1,a(m)*x^m/m!)+O(x^(n+3)));if(n<3,n!*polcoeff(A,n),n!*polcoeff(subst(A,x,A)^2/A^2,n-1)/(n-2))}
%Y A179497 Cf. A179498, A179499, A179420.
%K A179497 nonn
%O A179497 1,2
%A A179497 _Paul D. Hanna_, Jul 31 2010