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A219269 G.f. satisfies: A(x) = Sum_{n>=0} x^n/A(x)^n * Product_{k=0..n} k!.

%I A219269 #14 Nov 16 2012 23:24:18
%S A219269 1,1,1,8,247,33184,24678266,125237615376,5055581949347115,
%T A219269 1834887966372111613136,6658588234946979374670842054,
%U A219269 265790194051800257952649093995518288,127313960109916568757252293587045497552163302,792786695940715289991550398242378268738388375150573312
%N A219269 G.f. satisfies: A(x) = Sum_{n>=0} x^n/A(x)^n * Product_{k=0..n} k!.
%F A219269 G.f. A(x) satisfies: [x^n] A(x)^(n+1)/(n+1) = Product_{k=0..n} k! = superfactorial A000178(n).
%F A219269 G.f.: x / Series_Reversion(x*F(x)) where F(x) = Sum_{n>=0} x^n*Product_{k=0..n} k! is the g.f. of A000178.
%e A219269 G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 247*x^4 + 33184*x^5 + 24678266*x^6 +...
%e A219269 Given g.f. A(x), the table of coefficients in A(x)^n begins:
%e A219269 n=1: [(1), 1, 1, 8, 247, 33184, 24678266, 125237615376, ...];
%e A219269 n=2: [1,(2), 3, 18, 511, 66878, 49423458, 250524657604, ...];
%e A219269 n=3: [1, 3, (6), 31, 795, 101109, 74236366, 375861227934, ...];
%e A219269 n=4: [1, 4, 10,(48), 1103, 135908, 99117818, 501247428704, ...];
%e A219269 n=5: [1, 5, 15, 70,(1440), 171311, 124068685, 626683363390, ...];
%e A219269 n=6: [1, 6, 21, 98, 1812,(207360), 149089887, 752169136662, ...];
%e A219269 n=7: [1, 7, 28, 133, 2226, 244104,(174182400), 877704854447, ...];
%e A219269 n=8: [1, 8, 36, 176, 2690, 281600, 199347264,(1003290624000), ...]; ...
%e A219269 in which the main diagonal generates the superfactorials (A000178):
%e A219269 [1/1, 2/2, 6/3, 48/4, 1440/5, 207360/6, 174182400/7, 1003290624000/8, ...].
%e A219269 The logarithmic derivative of the superfactorials forms another diagonal:
%e A219269 A219266 = [1, 3, 31, 1103, 171311, 149089887, 877704854447, ...].
%o A219269 (PARI) {a(n)=polcoeff(x/serreverse(x*sum(k=0,n+1,prod(j=0,k,j!)*x^k)+x^2*O(x^n)),n)}
%o A219269 for(n=0,21,print1(a(n),", "))
%Y A219269 Cf. A219266, A000178, A219270.
%K A219269 nonn
%O A219269 0,4
%A A219269 _Paul D. Hanna_, Nov 16 2012