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A234296 E.g.f.: 1 + Integral (1 + Integral (1 + Integral (1 + Integral (1 + ...)^16 dx)^8 dx)^4 dx)^2 dx.

%I A234296 #10 Jan 05 2014 11:24:33
%S A234296 1,1,2,10,112,2544,110944,9088160,1395985024,405640228736,
%T A234296 225812739686144,243825339649539840,515865727833142919168,
%U A234296 2154502537039937189822464,17852312368540223401725132800,294428418578798287467609655705600,9684259826489059207872454620228222976
%N A234296 E.g.f.: 1 + Integral (1 + Integral (1 + Integral (1 + Integral (1 + ...)^16 dx)^8 dx)^4 dx)^2 dx.
%H A234296 Vaclav Kotesovec, <a href="/A234296/b234296.txt">Table of n, a(n) for n = 0..80</a>
%e A234296 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 10*x^3/3! + 112*x^4/4! + 2544*x^5/5! +...
%e A234296 such that
%e A234296 A(x) = 1 + Integral B(x)^2 dx,
%e A234296 B(x) = 1 + Integral C(x)^4 dx,
%e A234296 C(x) = 1 + Integral D(x)^8 dx,
%e A234296 D(x) = 1 + Integral E(x)^16 dx,
%e A234296 E(x) = 1 + Integral F(x)^32 dx,
%e A234296 F(x) = 1 + Integral G(x)^64 dx, ...
%e A234296 The coefficients in these series begin:
%e A234296 A: [1, 1, 2, 10, 112, 2544, 110944, 9088160, 1395985024, ...];
%e A234296 B: [1, 1, 4, 44, 1048, 48472, 4171008, 663109888, 196890206720, ...];
%e A234296 C: [1, 1, 8, 184, 9040, 845712, 144855616, 45401856704, ...];
%e A234296 D: [1, 1, 16, 752, 75040, 14126752, 4830297984, 3006883867264, ...];
%e A234296 E: [1, 1, 32, 3040, 611392, 230931264, 157795465984, ...];
%e A234296 F: [1, 1, 64, 12224, 4935808, 3734695552, 5101948036608, ...];
%e A234296 G: [1, 1, 128, 49024, 39665920, 60075785472, 164109335366656, ...];
%e A234296 H: [1, 1, 256, 196352, 318046720, 963787028992, 5265107899521024, ...]; ...
%e A234296 To illustrate a(n) = d^n/dx^n A(x) at x=0, take successive derivatives of A=A(x):
%e A234296 A' = B^2;
%e A234296 A'' = 2*B*C^4;
%e A234296 A''' = 2*C^8 + 8*B*C^3*D^8;
%e A234296 A'''' = 24*C^7*D^8 + 24*B*C^2*D^16 + 64*B*C^3*D^7*E^16; ...
%e A234296 and then evaluate at x=0, where 1=A(0)=B(0)=C(0)=D(0)=E(0)=...
%o A234296 (PARI) {a(n)=local(A=1); for(k=0, n-1, A=1+intformal((A+x*O(x^n))^(2^(n-k)))); n!*polcoeff(A, n)}
%o A234296 for(n=0, 20, print1(a(n), ", "))
%Y A234296 Cf. A120959, A234301.
%K A234296 nonn
%O A234296 0,3
%A A234296 _Paul D. Hanna_, Dec 24 2013