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A340943
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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)).
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5
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1, 1, 5, 45, 482, 5665, 70725, 921174, 12379878, 170435921, 2391736448, 34089385297, 492181254691, 7183748957321, 105830560089572, 1571662656809121, 23504719106546214, 353701665355036178, 5351873694519004045, 81378581395212130011
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OFFSET
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0,3
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COMMENTS
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Equals row k = 4 of rectangular table A340940.
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LINKS
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FORMULA
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Given g.f. A(x), the following sums are all equal:
(1) B(x) = Sum_{n>=0} x^n*A(x)^(4*n)/(1 - x*A(x)^n),
(2) B(x) = Sum_{n>=0} x^n*A(x)^(3*n)/(1 - x*A(x)^(4*n+1)),
(3) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(4*n+3)),
(4) B(x) = Sum_{n>=0} x^n/(1 - x*A(x)^(n+4)),
(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))),
(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)));
see the example section for the value of B(x).
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EXAMPLE
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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 + ...
such that
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) + ...
and
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) + ...
also
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) + ...
further,
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) + ...
where
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 + ...
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PROG
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(PARI) {a(n) = my(A=[1, 1]); for(i=1, n, A=concat(A, 0); H=A; A=concat(A, 0);
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] }
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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