cp's OEIS Frontend

A101189 G.f. A(x) is defined as the limit A(x) = lim_{n->oo} F(n)^(1/2^(n-1)) where F(n) is defined by F(n) = F(n-1)^2 + (2*x)^(2^n-1) for n >= 1 with F(0) = 1.

%I A101189 #10 Mar 05 2024 11:25:49
%S A101189 1,2,0,4,-8,16,-40,144,-512,1696,-5696,19840,-70048,247744,-880128,
%T A101189 3152768,-11386624,41389568,-151273728,555794944,-2052141056,
%U A101189 7610274816,-28331018240,105833345024,-396594444800,1490425179136,-5615651143680,21209004267520,-80276663808000
%N A101189 G.f. A(x) is defined as the limit A(x) = lim_{n->oo} F(n)^(1/2^(n-1)) where F(n) is defined by F(n) = F(n-1)^2 + (2*x)^(2^n-1) for n >= 1 with F(0) = 1.
%C A101189 Sequences A101190 and A101191 are related to doubly exponential numbers A003095 and to Catalan numbers (A000108).
%H A101189 Paul D. Hanna, <a href="/A101189/b101189.txt">Table of n, a(n) for n = 0..1025</a>
%F A101189 G.f. A(x) = ( Sum_{n>=0} A101190(n)/2^A005187(n) * (2*x)^n )^2.
%F A101189 G.f. A(x) = ( Sum_{n>=0} A101191(n)/2^A004134(n) * (2*x)^n )^4.
%e A101189 G.f.: A(x) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 144*x^7 - 512*x^8 + 1696*x^9 - 5696*x^10 + 19840*x^11 - 70048*x^12 + ...
%e A101189 GENERATING METHOD.
%e A101189 We can illustrate the generating method for g.f. A(x) as follows.
%e A101189 Given F(n) = F(n-1)^2 + (2*x)^(2^n-1) for n >= 1 with F(0) = 1,
%e A101189 the first few polynomials generated by F(n) begin
%e A101189 F(0) = 1,
%e A101189 F(1) = F(0)^2 + (2*x)^(2^1-1) = 1 + 2*x,
%e A101189 F(2) = F(1)^2 + (2*x)^(2^2-1) = 1 + 4*x + 4*x^2 + 8*x^3,
%e A101189 F(3) = F(2)^2 + (2*x)^(2^3-1) = 1 + 8*x + 24*x^2 + 48*x^3 + 80*x^4 + 64*x^5 + 64*x^6 + 128*x^7,
%e A101189 F(4) = F(3)^2 + (2*x)^(2^4-1) = = 1 + 16*x + 112*x^2 + 480*x^3 + 1504*x^4 + 3712*x^5 + 7296*x^6 + 12032*x^7 + 17664*x^8 + 22528*x^9 + 26624*x^10 + 28672*x^11 + 20480*x^12 + 16384*x^13 + 16384*x^14 + 32768*x^15,
%e A101189 ...
%e A101189 and the 2^(n-1)-th root of F(n) yields the series shown by
%e A101189 F(1)^(1/2^0) = 1 + 2*x,
%e A101189 F(2)^(1/2^1) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 112*x^7 - 320*x^8 + 928*x^9 - 2752*x^10 + 8320*x^11 - 25504*x^12 + ...,
%e A101189 F(3)^(1/2^2) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 144*x^7 - 512*x^8 + 1696*x^9 - 5696*x^10 + 19840*x^11 - 70048*x^12 + ...,
%e A101189 F(4)^(1/2^3) = 1 + 2*x + 4*x^3 - 8*x^4 + 16*x^5 - 40*x^6 + 144*x^7 - 512*x^8 + 1696*x^9 - 5696*x^10 + 19840*x^11 - 70048*x^12 + ...,
%e A101189 ...
%e A101189 The limit of this process tends to the g.f. A(x).
%o A101189 (PARI) {a(n) = my(F=1,A,L); if(n==0,A=1, L = ceil(log(n+1)/log(2)); for(k=1,L, F = F^2 + (2*x)^(2^k-1)  +x*O(x^n)); A = polcoeff(F^(1/(2^(L-1))),n)); A}
%o A101189 for(n=0,32, print1(a(n),", "))
%Y A101189 Cf. A101190, A101191, A005187, A004134, A003095.
%K A101189 sign
%O A101189 0,2
%A A101189 _Paul D. Hanna_, Dec 03 2004
%E A101189 Entry revised by _Paul D. Hanna_, Mar 05 2024