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%I #13 Dec 15 2015 23:42:53
%S 1,1,2,4,10,20,48,120,320,640,1520,3648,9216,23040,59520,158720,
%T 426240,852480,2037760,4839680,12038400,28892160,72142848,182255616,
%U 464781312,1161953280,2957967360,7641415680,19777781760,52740751360,138779688960,372690616320,990786355200,1981572710400,4793092669440,11457362657280,28548724162560
%N Terms satisfy: a(2*n) = a(n)*b(n) and a(2*n+1) = a(n+1)*b(n) for n>=0 with a(0)=1, where A(x)^2 = Sum_{n>=0} b(n)*x^n and g.f. A(x) = Sum_{n>=0} a(n)*x^n.
%C What is the limit a(n)^(1/n) = ? The limit is at least 2.535...
%H Paul D. Hanna, <a href="/A265264/b265264.txt">Table of n, a(n) for n = 0..3000</a>
%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 20*x^5 + 48*x^6 + 120*x^7 + 320*x^8 + 640*x^9 + 1520*x^10 + 3648*x^11 + 9216*x^12 +...
%e where
%e A(x)^2 = 1 + 2*x + 5*x^2 + 12*x^3 + 32*x^4 + 76*x^5 + 192*x^6 + 496*x^7 + 1332*x^8 + 3184*x^9 + 7920*x^10 + 19776*x^11 + 50432*x^12 +...+ A265265(n)*x^n +...
%e Illustration of initial terms, both a(n) of A(x) and b(n) of A(x)^2:
%e a(0) = a(0)*b(0) = 1; b(0) = 1;
%e a(1) = a(1)*b(0) = 1; b(1) = 2 = 1*1 + 1*1;
%e a(2) = a(1)*b(1) = 2; b(2) = 5 = 1*2 + 1*1 + 2*1;
%e a(3) = a(2)*b(1) = 4; b(3) = 12 = 1*4 + 1*2 + 2*1 + 4*1;
%e a(4) = a(2)*b(2) = 10; b(4) = 32 = 1*10 + 1*4 + 2*2 + 4*1 + 10*1;
%e a(5) = a(3)*b(2) = 28; b(5) = 76;
%e a(6) = a(3)*b(3) = 48; b(6) = 192;
%e a(7) = a(4)*b(3) = 120; b(7) = 496;
%e a(8) = a(4)*b(4) = 320; b(8) = 1332; ...
%o (PARI) {a(n) = my(A=1+x); for(k=2, n, A = A + a((k+1)\2) * polcoeff(A^2, k\2) * x^k +x*O(x^n) ); polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[1, 1]); for(k=2, n, A = concat(A, A[(k+1)\2+1]*Vec(Ser(A)^2)[k\2+1]) ); A[n+1]}
%o for(n=0, 40, print1(a(n), ", "))
%o (PARI) /* Generates N terms rather quickly: */
%o N=300; A=[1, 1]; for(k=2, N, A = concat(A, A[(k+1)\2+1]*Vec(Ser(A)^2)[k\2+1]) ); A
%Y Cf. A265265 (A^2), A257889.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 15 2015
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