A340941 - OEIS

%I #27 Mar 08 2021 12:06:06

%S 1,1,1,3,10,36,137,545,2244,9500,41151,181736,816085,3718508,17165117,

%T 80172679,378513020,1804991424,8688578508,42199027520,206723209249,

%U 1021191612686,5086213545738,25540278088827,129302804317291,660055084179260,3397896035465901

%N G.f. A(x) satisfies: A(x) = (A(x) - x)*(1 - x*A(x)) * Sum_{n>=0} x^n/(1 - x*A(x)^n).

%C Equals row k = 0 of rectangular table A340940.

%F G.f. A(x) satisfies:

%F (1) Sum_{n>=0} x^n/(1 - x*A(x)^n)  =  A(x)/((A(x) - x)*(1 - x*A(x))).

%F (2) Sum_{n>=0} x^(2*n) * A(x)^(n^2) * (1 + x*A(x)^n) / (1 - x*A(x)^n))  =  A(x)/((A(x) - x)*(1 - x*A(x))).

%e G.f. A(x) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 36*x^5 + 137*x^6 + 545*x^7 + 2244*x^8 + 9500*x^9 + 41151*x^10 + 181736*x^11 + 816085*x^12 + ...

%e such that

%e A(x)/((A(x) - x)*(1 - x*A(x)))  =  1/(1-x) + x/(1 - x*A(x)) + x^2/(1 - x*A(x)^2) + x^3/(1 - x*A(x)^3) + x^4/(1 - x*A(x)^4) + x^5/(1 - x*A(x)^5) + ...

%e also

%e A(x)/((A(x) - x)*(1 - x*A(x)))  =  (1+x)/(1-x) + x^2*A(x)*(1 + x*A(x))/(1 - x*A(x)) + x^4*A(x)^4*(1 + x*A(x)^2)/(1 - x*A(x)^2) + x^6*A(x)^9*(1 + x*A(x)^3)/(1 - x*A(x)^3) + x^8*A(x)^16*(1 + x*A(x)^4)/(1 - x*A(x)^4) + ...

%e where

%e A(x)/((A(x) - x)*(1 - x*A(x)))  =  1 + 2*x + 3*x^2 + 5*x^3 + 10*x^4 + 25*x^5 + 75*x^6 + 255*x^7 + 940*x^8 + 3660*x^9 + 14827*x^10 + 61927*x^11 + ...

%e GENERATING METHOD.

%e Note that iteration of the following equation does not converge to the g.f. A(x): A(x) = (A(x) - x)*(1 - x*A(x)) * Sum_{n>=0} x^n/(1 - x*A(x)^n); however, the coefficients of A(x) may be determined by the method described below.

%e Let P(x) be a partial sum of the power series A(x), then the next term in the series can be determined as follows.

%e (1) Start with P(x) = 1 + x, then the coefficient a(2)=1 of x^2 in A(x) appears as the coefficient of x^4 in (P(x)-x)*(1-x*P(x))*Sum_{n=0..4} x^n/(1 - x*P(x)^n) = 1 + x + 0*x^2 + 0*x^3 + 1*x^4 + ...

%e (2) Set P(x) = 1 + x + x^2, then the coefficient a(3)=3 of x^3 in A(x) appears as the coefficient of x^5 in (P(x)-x)*(1-x*P(x))*Sum_{n=0..5} x^n/(1 - x*P(x)^n) = 1 + x + x^2 + 0*x^3 + 0*x^4 + 3*x^5 + ...

%e (3) Set P(x) = 1 + x + x^2 + 3*x^3, then the coefficient a(4)=10 of x^4 in A(x) appears as the coefficient of x^6 in (P(x)-x)*(1-x*P(x))*Sum_{n=0..6} x^n/(1 - x*P(x)^n) = 1 + x + x^2 + 3*x^3 + 0*x^4 + 0*x^5 + 10*x^6 + ...

%e (4) Set P(x) = 1 + x + x^2 + 3*x^3 + 10*x^4, then the coefficient a(5)=36 of x^5 in A(x) appears as the coefficient of x^7 in (P(x)-x)*(1-x*P(x))*Sum_{n=0..7} x^n/(1 - x*P(x)^n) = 1 + x + x^2 + 3*x^3 + 10*x^4 + 0*x^5 + 0*x^6 + 36*x^7 + ...

%e Continue in this way to uniquely determine all the coefficients in g.f. A(x).

%o (PARI) {a(n) = my(A=1+x +x^3*O(x^n),H=A);

%o for(k=1, n, A = (A-x)*(1-x*A) * sum(m=0, n+3, x^m / (1 - x*A^m +x^3*O(x^n)) );

%o A = truncate( H + polcoeff(A, k+2)*x^k ) +x^3*O(x^n); H=A);polcoeff(A,n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A340940, A340942, A340894, A340895..

%K nonn

%O 0,4

%A _Paul D. Hanna_, Feb 03 2021