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A159314 Rectangular array, read by antidiagonals, where row e.g.f.s, R(n,x), satisfy: d/dx log( R(n,x) ) = R(n+1,x)^(2^n) with R(n,0) = 1; that is, the logarithmic derivative of the e.g.f. of row n equals the e.g.f. of row n+1 to the 2^n power, for n>=0.

%I A159314 #5 Apr 20 2013 03:42:49
%S A159314 1,1,1,1,1,2,1,1,3,7,1,1,5,19,41,1,1,9,61,225,406,1,1,17,217,1481,
%T A159314 4801,7127,1,1,33,817,10737,66361,185523,235147,1,1,65,3169,81761,
%U A159314 988561,5390285,13298659,15191966,1,1,129,12481,638145,15269281,164637369
%N A159314 Rectangular array, read by antidiagonals, where row e.g.f.s, R(n,x), satisfy: d/dx log( R(n,x) ) = R(n+1,x)^(2^n) with R(n,0) = 1; that is, the logarithmic derivative of the e.g.f. of row n equals the e.g.f. of row n+1 to the 2^n power, for n>=0.
%F A159314 T(n,k) = Sum_{i=0..k-1} C(k-1,i)*2^(n*i)*T(1,i)*T(n,k-1-i) for k>0 with T(n,0)=1, for n>=0.
%F A159314 Row e.g.f.s, R(n,x), satisfy:
%F A159314 (1) R'(n,x)/R(n,x) = R(n+1,x)^(2^n) with R(n,0) = 1;
%F A159314 (2) R(n,x) = R(n+m,x/2^m)^(2^m) for m >= -n.
%e A159314 Array begins:
%e A159314 1,1,2,7,41,406,7127,235147,15191966,1953128401,501361942127,...;
%e A159314 1,1,3,19,225,4801,185523,13298659,1815718305,481790947681,...;
%e A159314 1,1,5,61,1481,66361,5390285,803252341,224927827601,...;
%e A159314 1,1,9,217,10737,988561,164637369,49987302697,28333326990177,...;
%e A159314 1,1,17,817,81761,15269281,5149256177,3155353490257,...;
%e A159314 1,1,33,3169,638145,240072001,162919458273,200565037419169,...;
%e A159314 1,1,65,12481,5042561,3807826561,5184101454785,12792473234253121,...;
%e A159314 1,1,129,49537,40092417,60660860161,165425163421569,...;
%e A159314 1,1,257,197377,319751681,968467745281,5286172203486977,...;
%e A159314 1,1,513,787969,2554072065,15478671283201,169038775947894273,...;
%e A159314 1,1,1025,3148801,20416829441,247524381173761,5407342625815542785,...;
%e A159314 ...
%e A159314 where row e.g.f.s begin:
%e A159314 R(0,x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 41*x^4/4! + 406*x^5/5! +...;
%e A159314 R(1,x) = 1 + x + 3*x^2/2! +19*x^3/3! +225*x^4/4! +4801*x^5/5! +...;
%e A159314 R(2,x) = 1 + x + 5*x^2/2! +61*x^3/3!+1481*x^4/4!+66361*x^5/5! +...;
%e A159314 ...
%e A159314 Row e.g.f.s satisfy: R(n+1,x)^(2^n) = d/dx log( R(n,x) ):
%e A159314 R(1,x)^1 = d/dx log(1+x +2*x^2/2! +7*x^3/3! +41*x^4/4! +...);
%e A159314 R(2,x)^2 = d/dx log(1+x +3*x^2/2! +19*x^3/3! +225*x^4/4! +...);
%e A159314 R(3,x)^4 = d/dx log(1+x +5*x^2/2! +61*x^3/3! +1481*x^4/4! +...);
%e A159314 R(4,x)^8 = d/dx log(1+x +9*x^2/2! +217*x^3/3! +10737*x^4/4! +...);
%e A159314 ...
%e A159314 Examples of R(n,x) = R(n+m,x/2^m)^(2^m):
%e A159314 R(n-1,x) = R(n,x/2)^2 and R(n+1,x) = R(n,2x)^(1/2);
%e A159314 R(0,x) = R(n,x/2^n)^(2^n) and R(n,x) = R(0,2^n*x)^(1/2^n).
%o A159314 (PARI) {T(n,k)=if(k==0,1,sum(i=0,k-1,2^(n*i)*binomial(k-1,i)*T(1,i)*T(n,k-1-i)))}
%o A159314 (PARI) {T(n, k)=local(A=vector(n+k+2, j, 1+j*x)); for(i=0, n+k+1, for(j=0, n+k, m=n+k+1-j; A[m]=exp(intformal((A[m+1]+x*O(x^k))^(2^(m-1)))))); k!*polcoeff(A[n+1], k, x)}
%Y A159314 Cf. rows: A159315, A126444, A159316, diagonal: A159317, variant: A145085.
%K A159314 nonn,tabl
%O A159314 0,6
%A A159314 _Paul D. Hanna_, Apr 19 2009