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%I #9 Feb 19 2022 01:41:22
%S 1,1,5,43,443,5009,60104,751778,9696036,128037209,1722632206,
%T 23530913551,325494250943,4550333846746,64189733915195,
%U 912589001283146,13062908562155459,188107110626083146,2723185267618504739,39610394334267885677
%N G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(n+4)) = Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^(3*n+3)).
%F Given g.f. A(x), the following sums are all equal.
%F (1) B(x) = Sum_{n>=0} x^n*A(x)^(1*n) / (1 - x*A(x)^(1*n+4));
%F (2) B(x) = Sum_{n>=0} x^n*A(x)^(2*n) / (1 - x*A(x)^(3*n+3));
%F (3) B(x) = Sum_{n>=0} x^n*A(x)^(3*n) / (1 - x*A(x)^(3*n+2));
%F (4) B(x) = Sum_{n>=0} x^n*A(x)^(4*n) / (1 - x*A(x)^(1*n+1));
%F (5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+5*n) * (1 - x^2*A(x)^(2*n+5)) / ((1 - x*A(x)^(n+1))*(1 - x*A(x)^(n+4))),
%F (6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(3*n^2+5*n) * (1 - x^2*A(x)^(6*n+5)) / ((1 - x*A(x)^(3*n+2))*(1 - x*A(x)^(3*n+3)));
%F see the example section for the value of B(x).
%e G.f.: A(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 + ...
%e such that the following sums are all equal:
%e (1) B(x) = 1/(1 - x*A(x)^4) + x*A(x)^1/(1 - x*A(x)^5) + x^2*A(x)^2/(1 - x*A(x)^6) + x^3*A(x)^3/(1 - x*A(x)^7) + x^4*A(x)^4/(1 - x*A(x)^8) + ...
%e (2) B(x) = 1/(1 - x*A(x)^3) + x*A(x)^2/(1 - x*A(x)^6) + x^2*A(x)^4/(1 - x*A(x)^9) + x^3*A(x)^6/(1 - x*A(x)^12) + x^4*A(x)^8/(1 - x*A(x)^15) + ...
%e (3) B(x) = 1/(1 - x*A(x)^2) + x*A(x)^3/(1 - x*A(x)^5) + x^2*A(x)^6/(1 - x*A(x)^8) + x^3*A(x)^9/(1 - x*A(x)^11) + x^4*A(x)^12/(1 - x*A(x)^14) + ...
%e (4) B(x) = 1/(1 - x*A(x)^1) + x*A(x)^4/(1 - x*A(x)^2) + x^2*A(x)^8/(1 - x*A(x)^3) + x^3*A(x)^12/(1 - x*A(x)^4) + x^4*A(x)^16/(1 - x*A(x)^5) + ...
%e where
%e B(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 + 17575262387*x^11 + 240012293969*x^12 + ... (see A351772).
%o (PARI) {a(n) = my(A=[1,1,0]); for(i=0,n, A=concat(A,0);
%o B1 = sum(m=0,#A, x^m*Ser(A)^(2*m)/(1 - x*Ser(A)^(3*m+3)) );
%o B2 = sum(m=0,#A, x^m*Ser(A)^(4*m)/(1 - x*Ser(A)^(1*m+1)) );
%o A[#A-1] = polcoeff((B1 - B2)/2, #A);); A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = my(A=[1,1,0]); for(i=0,n, A=concat(A,0);
%o B1 = sum(m=0,#A, x^m*Ser(A)^(3*m)/(1 - x*Ser(A)^(3*m+2)) );
%o B2 = sum(m=0,#A, x^m*Ser(A)^(1*m)/(1 - x*Ser(A)^(1*m+4)) );
%o A[#A-1] = polcoeff((B1 - B2)/2, #A);); A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A340895, A340943, A351772.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 14 2021
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