|
|
FORMULA
|
The g.f. A(x) of this sequence can be determined from the g.f. F(x) of A350117 as follows.
(1) A(x) = Sum_{n>=0} x^n*F(x)^(1*n) / (1 - x*F(x)^(1*n+4));
(2) A(x) = Sum_{n>=0} x^n*F(x)^(2*n) / (1 - x*F(x)^(3*n+3));
(3) A(x) = Sum_{n>=0} x^n*F(x)^(3*n) / (1 - x*F(x)^(3*n+2));
(4) A(x) = Sum_{n>=0} x^n*F(x)^(4*n) / (1 - x*F(x)^(1*n+1));
(5) A(x) = Sum_{n>=0} x^(2*n) * F(x)^(n^2+5*n) * (1 - x^2*F(x)^(2*n+5)) / ((1 - x*F(x)^(n+1))*(1 - x*F(x)^(n+4))),
(6) A(x) = Sum_{n>=0} x^(2*n) * F(x)^(3*n^2+5*n) * (1 - x^2*F(x)^(6*n+5)) / ((1 - x*F(x)^(3*n+2))*(1 - x*F(x)^(3*n+3)));
see the example section for the series expansion of F(x).
|
|
|
EXAMPLE
|
G.f.: A(x) = 1 + 2*x + 8*x^2 + 51*x^3 + 442*x^4 + 4534*x^5 + 51182*x^6 + 613806*x^7 + 7675397*x^8 + 98971497*x^9 + 1306630823*x^10 + ...
such that
(1) A(x) = 1/(1 - x*F(x)^4) + x*F(x)^1/(1 - x*F(x)^5) + x^2*F(x)^2/(1 - x*F(x)^6) + x^3*F(x)^3/(1 - x*F(x)^7) + x^4*F(x)^4/(1 - x*F(x)^8) + ...
(2) A(x) = 1/(1 - x*F(x)^3) + x*F(x)^2/(1 - x*F(x)^6) + x^2*F(x)^4/(1 - x*F(x)^9) + x^3*F(x)^6/(1 - x*F(x)^12) + x^4*F(x)^8/(1 - x*F(x)^15) + ...
(3) A(x) = 1/(1 - x*F(x)^2) + x*F(x)^3/(1 - x*F(x)^5) + x^2*F(x)^6/(1 - x*F(x)^8) + x^3*F(x)^9/(1 - x*F(x)^11) + x^4*F(x)^12/(1 - x*F(x)^14) + ...
(4) A(x) = 1/(1 - x*F(x)^1) + x*F(x)^4/(1 - x*F(x)^2) + x^2*F(x)^8/(1 - x*F(x)^3) + x^3*F(x)^12/(1 - x*F(x)^4) + x^4*F(x)^16/(1 - x*F(x)^5) + ...
where
F(x) = 1 + x + 5*x^2 + 43*x^3 + 443*x^4 + 5009*x^5 + 60104*x^6 + 751778*x^7 + 9696036*x^8 + 128037209*x^9 + 1722632206*x^10 + ... + A350117(n)*x^n + ...
|
|
|
PROG
|
(PARI) {a(n) = my(F=[1, 1, 0]); for(i=0, n, F=concat(F, 0);
A1 = sum(m=0, #F, x^m*Ser(F)^(2*m)/(1 - x*Ser(F)^(3*m+3)) );
A2 = sum(m=0, #F, x^m*Ser(F)^(4*m)/(1 - x*Ser(F)^(1*m+1)) );
F[#F-1] = polcoeff((A1 - A2)/2, #F); ); polcoeff(A1, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(F=[1, 1, 0]); for(i=0, n, F=concat(F, 0);
A1 = sum(m=0, #F, x^m*Ser(F)^(3*m)/(1 - x*Ser(F)^(3*m+2)) );
A2 = sum(m=0, #F, x^m*Ser(F)^(1*m)/(1 - x*Ser(F)^(1*m+4)) );
F[#F-1] = polcoeff((A1 - A2)/2, #F); ); polcoeff(A1, n)}
for(n=0, 30, print1(a(n), ", "))
|
