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%I #19 Mar 08 2021 12:10:10
%S 1,1,5,45,482,5665,70725,921174,12379878,170435921,2391736448,
%T 34089385297,492181254691,7183748957321,105830560089572,
%U 1571662656809121,23504719106546214,353701665355036178,5351873694519004045,81378581395212130011
%N G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(4*n)/(1 - x*A(x)^n) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(4*n+3)).
%C Equals row k = 4 of rectangular table A340940.
%F Given g.f. A(x), the following sums are all equal:
%F (1) B(x) = Sum_{n>=0} x^n*A(x)^(4*n)/(1 - x*A(x)^n),
%F (2) B(x) = Sum_{n>=0} x^n*A(x)^(3*n)/(1 - x*A(x)^(4*n+1)),
%F (3) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(4*n+3)),
%F (4) B(x) = Sum_{n>=0} x^n/(1 - x*A(x)^(n+4)),
%F (5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+4*n) * (1 - x^2*A(x)^(2*n+4)) / ((1 - x*A(x)^n)*(1 - x*A(x)^(n+4))),
%F (6) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(4*n^2+4*n) * (1 - x^2*A(x)^(8*n+4)) / ((1 - x*A(x)^(4*n+1))*(1 - x*A(x)^(4*n+3)));
%F see the example section for the value of B(x).
%e G.f.: A(x) = 1 + x + 5*x^2 + 45*x^3 + 482*x^4 + 5665*x^5 + 70725*x^6 + 921174*x^7 + 12379878*x^8 + 170435921*x^9 + 2391736448*x^10 + ...
%e such that
%e B(x) = 1/(1-x) + x*A(x)^4/(1 - x*A(x)) + x^2*A(x)^8/(1 - x*A(x)^2) + x^3*A(x)^12/(1 - x*A(x)^3) + x^4*A(x)^16/(1 - x*A(x)^4) + ...
%e and
%e B(x) = 1/(1 - x*A(x)) + x*A(x)^3/(1 - x*A(x)^5) + x^2*A(x)^6/(1 - x*A(x)^9) + x^3*A(x)^9/(1 - x*A(x)^13) + x^4*A(x)^12/(1 - x*A(x)^17) + ...
%e also
%e B(x) = 1/(1 - x*A(x)^3) + x*A(x)/(1 - x*A(x)^7) + x^2*A(x)^2/(1 - x*A(x)^11) + x^3*A(x)^3/(1 - x*A(x)^15) + x^4*A(x)^4/(1 - x*A(x)^19) + ...
%e further,
%e B(x) = 1/(1 - x*A(x)^4) + x/(1 - x*A(x)^5) + x^2/(1 - x*A(x)^6) + x^3/(1 - x*A(x)^7) + x^4/(1 - x*A(x)^8) + ...
%e where
%e B(x) = 1 + 2*x + 7*x^2 + 43*x^3 + 380*x^4 + 4032*x^5 + 47234*x^6 + 588683*x^7 + 7657593*x^8 + 102796547*x^9 + 1413743374*x^10 + ...
%o (PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); H=A; A=concat(A, 0);
%o H[#A-1] = -polcoeff( sum(m=0, #A, x^m/(1 - x*Ser(A)^(m+4)) ) - sum(m=0, #A, x^m*Ser(A)^m/(1 - x*Ser(A)^(4*m+3)) ), #A)/3; A=H); A[n+1] }
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A340940, A340941, A340942, A340894, A340895, A341376.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 04 2021
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