|
%I #11 Feb 04 2023 09:10:58
%S 1,1,4,15,63,295,1502,8167,46873,281672,1761798,11418480,76415644,
%T 526594846,3728435747,27073765165,201325681384,1531247489953,
%U 11899881220174,94409837555587,764105555574024,6304959856949278
%N G.f.: A(x) = R_2(x)/R_1(x), where R_2(x) and R_1(x) are the g.f.s of row 2 (A124542) and row 1 (A124531), respectively, of table A124540.
%C In table A124540, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = [ Sum_{k>=0} y^k*R_k(y)^n ]^n for n>=0.
%e G.f.: A(x) = R_2(x)/R_1(x), where row g.f.s are:
%e R_2(x) = 1 + 2x + 7x^2 + 26x^3 + 107x^4 + 486x^5 + 2398x^6 + ... and
%e R_1(x) = 1 + x + 2x^2 + 5x^3 + 16x^4 + 62x^5 + 274x^6 + ..., so that
%e A(x) = 1 + x + 4*x^2 + 15*x^3 + 63*x^4 + 295*x^5 + 1502*x^6 + ...
%o (PARI) {a(n)=local(R);R=vector(n+3,r,vector(n+3,c,1)); for(i=0,n+2,for(r=0,n+2,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^(r*c)+O(x^(n+1)))))); Vec(Ser(R[3])^2/Ser(R[2])+O(x^(n+1)))[n+1]}
%Y Cf. A124540 (table); rows: A124531, A124542, A124543, A124544, A124545, A124546.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Nov 05 2006
|
