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%I #50 Jan 05 2023 04:01:37
%S 1,1,2,5,10,28,70,170,340,960,2688,7308,18270,48440,117640,285600,
%T 571200,1543600,4358400,12038400,33707520,92875776,252506016,
%U 677656224,1694140560,4596220440,12186147680,32083749600,77917677600,203473437920,493981756800,1217092396800,2434184793600,6357547392000,17180514976000,46002675920000,129889908480000,349123704576000
%N Terms satisfy: a(2*n) = a(n)*b(n) and a(2*n+1) = a(n)*b(n+1) 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 5 divides a(3*2^n) for n>=0 (conjecture).
%C 5 does not divide a(3*2^n - 1) for n>=0 (conjecture).
%C What is the limit a(n)^(1/n) = ?
%H Paul D. Hanna, <a href="/A257889/b257889.txt">Table of n, a(n) for n = 0..2500</a>
%H Vaclav Kotesovec, <a href="/A257889/a257889.jpg">Plot of a(n+1)/a(n)</a>
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 28*x^5 + 70*x^6 + 170*x^7 + 340*x^8 + 960*x^9 + 2688*x^10 + 7308*x^11 + 18270*x^12 +...
%e where
%e A(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 34*x^4 + 96*x^5 + 261*x^6 + 692*x^7 + 1680*x^8 + 4540*x^9 + 12540*x^10 + 34552*x^11 + 92728*x^12 +...+ A265226(n)*x^n +...
%e Illustration of initial terms, both a(n) of A(x) and b(n) of A(x)^2:
%e a(0) = 1; b(0) = 1;
%e a(1) = 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(1)*b(2) = 5; b(3) = 14 = 1*5 + 1*2 + 2*1 + 5*1;
%e a(4) = a(2)*b(2) = 10; b(4) = 34 = 1*10 + 1*5 + 2*2 + 5*1 + 10*1;
%e a(5) = a(2)*b(3) = 28; b(5) = 96;
%e a(6) = a(3)*b(3) = 70; b(6) = 261;
%e a(7) = a(3)*b(4) = 170; b(7) = 692;
%e a(8) = a(4)*b(4) = 340; b(8) = 1680; ...
%o (PARI) {a(n) = my(A=1+x); for(k=2,n, A = A + a(k\2) * polcoeff(A^2, (k+1)\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\2+1]*Vec(Ser(A)^2)[(k+1)\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\2+1]*Vec(Ser(A)^2)[(k+1)\2+1]) ); A
%Y Cf. A265226 (A^2), A265264, A264927.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 13 2015
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