A378247 - OEIS

%I #7 Nov 21 2024 06:03:50

%S 1,1,3,10,39,161,699,3135,14427,67716,322959,1560585,7624007,37593476,

%T 186856061,935214523,4709265692,23841104525,121275951719,619558165489,

%U 3177346503440,16351749778167,84419824808865,437105510426235,2269266695980449,11810014285000263,61602685079710638

%N G.f. A(x) satisfies A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).

%C Compare to C(x)^2 = C( x^2 + 2*x*C(x)^2 ) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

%H Paul D. Hanna, <a href="/A378247/b378247.txt">Table of n, a(n) for n = 1..500</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas in which G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of A001190, the Wedderburn-Etherington numbers.

%F (1) A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).

%F (2) x = A( x - x^2 - x*G(x^2) ).

%F (3) x = A( x + x^2 - 2*x*G(x) + x*G(x)^2 ).

%F (4) x = A( x*sqrt(1 - 2*x^2 - G(x^4)) - x^2 ).

%F (5) x^2 = A( x^2*((1 - G(x))^2 + 2*x)^2 + x^4 ).

%F (6) G(A(x)) = 1 - sqrt(x/A(x) - A(x)).

%e G.f.: A(x) = x + x^2 + 3*x^3 + 10*x^4 + 39*x^5 + 161*x^6 + 699*x^7 + 3135*x^8 + 14427*x^9 + 67716*x^10 + 322959*x^11 + 1560585*x^12 + ...

%e where A(x)^2 = A( x^2 + 2*x*A(x)^2 + 2*A(x)^4 ).

%e RELATED SERIES.

%e Let G(x) be the g.f. of the Wedderburn-Etherington numbers, then

%e A( x - x^2 - x*G(x^2) ) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) begins

%e G(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + 23*x^8 + 46*x^9 + 98*x^10 + 207*x^11 + 451*x^12 + 983*x^13 + 2179*x^14 + ... + A001190(n)*x^n + ...

%e Let B(x) be the series reversion of g.f. A(x) so that B(A(x)) = x, then

%e B(x) = x - x^2 - x^3 - x^5 - x^7 - 2*x^9 - 3*x^11 - 6*x^13 - 11*x^15 - 23*x^17 - 46*x^19 - 98*x^21 - 207*x^23 + ...

%e where B(x) = x - x^2 - x*G(x^2).

%e SPECIFIC VALUES.

%e A(t) = 1/3 at t = 0.1804894059505127351310871071614416167035910065610113327...

%e   where 1/9 = A( t^2 + 2*t/9 + 2/81 ).

%e A(t) = 1/4 at t = 0.1708289565101545485579649480920097855916395263217351536...

%e   where 1/16 = A( t^2 + t/8 + 1/128 ).

%e A(t) = 1/5 at t = 0.1516661092515691718015998101146470241027491658579501286...

%e   where 1/25 = A( t^2 + 2*t/25 + 2/625 ).

%e A(t) = 1/6 at t = 0.1341268789797555579297424694390747929782019601987848246...

%e   where 1/36 = A( t^2 + t/18 + 1/648 ).

%e A(1/6) = 0.2368314953172156547771056118501694080205525703518284958...

%e A(1/7) = 0.1824082884402163049324182135107985537409785918465705698...

%e A(1/8) = 0.1515179821748020682616541846638756124979071552818869937...

%e A(1/9) = 0.1303577455916869424988611259176631850931169441135101392...

%e A(1/10) = 0.1146797533131163787803333792504789207692884367435306666...

%o (PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);

%o A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2 + 2*x*Ax^2 + 2*Ax^4) - Ax^2, #A) ); A[n+1]}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A001190, A374567, A271959, A271960, A374566.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Nov 20 2024