|
%I #5 Jan 10 2021 11:51:02
%S 1,5,9,20,50,102,365,1279,3894,13925,48903,169524,621128,2227855,
%T 7970244,29200251,107135216,395895525,1475549405,5498774091,
%U 20567632890,77314123025,291467136572,1103022636621,4187756743759,15934627999334,60781416980898,232363965399736
%N G.f. A(x) satisfies: A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x)^n)^3.
%C The g.f. A(x) of this sequence is motivated by the following identity:
%C Sum_{n>=0} C(t+n-1,n) * p^n/(1 - q*r^n)^s = Sum_{n>=0} C(s+n-1,n) * q^n/(1 - p*r^n)^t ;
%C here, p = x, q = x, r = x*A(x), s = 2, and t = 3.
%F G.f. A(x) satisfies the following relations.
%F (1) A(x) = Sum_{n>=0} (n+1) * x^n / (1 - x^(n+1)*A(x)^n)^3.
%F (2) A(x) = Sum_{n>=0} (n+1)*(n+2)/2 * x^n / (1 - x^(n+1)*A(x)^n)^2.
%e G.f.: A(x) = 1 + 5*x + 9*x^2 + 20*x^3 + 50*x^4 + 102*x^5 + 365*x^6 + 1279*x^7 + 3894*x^8 + 13925*x^9 + 48903*x^10 + ...
%e where
%e A(x) = 1/(1 - x)^3 + 2*x/(1 - x^2*A(x))^3 + 3*x^2/(1 - x^3*A(x)^2)^3 + 4*x^3/(1 - x^4*A(x)^3)^3 + 5*x^4/(1 - x^5*A(x)^4)^3 + ...
%e also
%e A(x) = 1/(1 - x)^2 + 3*x/(1 - x^2*A(x))^2 + 6*x^2/(1 - x^3*A(x)^2)^2 + 10*x^3/(1 - x^4*A(x)^3)^2 + 15*x^4/(1 - x^5*A(x)^4)^2 + ...
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, (m+1) * x^m / (1 - x^(m+1)*A^m +x*O(x^n))^3 )); polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = my(A=1); for(i=1,n, A = sum(m=0,n, (m+1)*(m+2)/2 * x^m / (1 - x^(m+1)*A^m +x*O(x^n))^2 )); ; polcoeff(A, n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A340329, A340338, A340355, A340356, A340357, A340358, A340359.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 07 2021
|
