|
%I #19 Mar 10 2024 00:20:39
%S 1,1,2,5,13,35,97,273,778,2240,6499,18976,55703,164243,486130,1443620,
%T 4299365,12836825,38413933,115184282,346005073,1041072108,3137060983,
%U 9465689545,28596915843,86492865522,261876842801,793661873276
%N G.f. A(x) satisfies x*A(x)^3 = B(x*A(x^3)) where B(x) = x/(1 - 3*x).
%C More generally, given A(x) satisfies x*A(x)^p = B(x*A(x^p)) where B(x) = x/(1-p*x), then it appears that A(x) is an integer series only when p is prime. This is a special case of sequences with g.f.s that satisfy the more general functional equation x*A(x)^m = B(x*A(x^m)) studied by Michael Somos; some other examples are A085748, A091188 and A091200.
%H Paul D. Hanna, <a href="/A091190/b091190.txt">Table of n, a(n) for n = 0..900</a>
%F From _Paul D. Hanna_, Mar 09 2024: (Start)
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F (1) A(x)^3 = A(x^3) / (1 - 3*x*A(x^3)).
%F (2) A(x) = x/Series_Reversion(D(x)) where D(x) = x*A(D(x)) is the g.f. of A370441.
%F (End)
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 35*x^5 + 97*x^6 + 273*x^7 + 778*x^8 + 2240*x^9 + 6499*x^10 + 18976*x^11 + 55703*x^12 + ...
%e where A(x)^3 = A(x^3) / (1 - 3*x*A(x^3)).
%e RELATED SERIES.
%e A(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 87*x^4 + 270*x^5 + 839*x^6 + 2607*x^7 + 8100*x^8 + 25169*x^9 + 78207*x^10 + 243009*x^11 + 755095*x^12 + ...
%e Also, D(x) = x*A(D(x)) is the g.f. of A370441, which begins
%e D(x) = x + x^2 + 3*x^3 + 12*x^4 + 54*x^5 + 261*x^6 + 1324*x^7 + 6952*x^8 + 37461*x^9 + ... + A370441(n)*x^n + ...
%e such that D(x)^3 = D( x^3 + 3*D(x)^4 ).
%t m = 28; B[x_] = x/(1 - 3 x); A[_] = 1;
%t Do[A[x_] = (B[x A[x^3]]/x)^(1/3) + O[x]^m // Normal, {m}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Oct 29 2019 *)
%o (PARI) {a(n) = my(A,p=3,m=1); if(n<0,0, m=1; A=1+O(x); while(m<=n, m*=p; A = x*subst(A,x,x^p); A = (A/(1-p*A)/x)^(1/p)); polcoeff(A,n))}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A085748, A091188, A091200, A370441.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 22 2004
%E Corrected by _T. D. Noe_, Oct 25 2006
| 