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A340894
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G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^n) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(2*n+1)).
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6
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1, 1, 3, 18, 126, 958, 7707, 64564, 557519, 4928784, 44398592, 406119900, 3762522607, 35236156779, 333052648607, 3173351871750, 30448930792460, 293980976342244, 2854070686906317, 27845872331417000, 272896298225616888, 2685318944226499424, 26521692874280137381
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OFFSET
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0,3
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COMMENTS
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Equals row k = 2 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)^(2*n)/(1 - x*A(x)^n),
(2) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(2*n+1)),
(3) B(x) = Sum_{n>=0} x^n/(1 - x*A(x)^(n+2)),
(4) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+2*n) * (1 - x^2*A(x)^(2*n+2)) / ((1 - x*A(x)^n)*(1 - x*A(x)^(n+2))),
(5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(2*n^2+2*n) * (1 + x*A(x)^(2*n+1)) / (1 - x*A(x)^(2*n+1));
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 + 3*x^2 + 18*x^3 + 126*x^4 + 958*x^5 + 7707*x^6 + 64564*x^7 + 557519*x^8 + 4928784*x^9 + 44398592*x^10 + 406119900*x^11 + ...
such that the following sums are all equal:
B(x) = 1/(1-x) + x*A(x)^2/(1 - x*A(x)) + x^2*A(x)^4/(1 - x*A(x)^2) + x^3*A(x)^6/(1 - x*A(x)^3) + x^4*A(x)^8/(1 - x*A(x)^4) + ...
and
B(x) = 1/(1-x*A(x)) + x*A(x)/(1 - x*A(x)^3) + x^2*A(x)^2/(1 - x*A(x)^5) + x^3*A(x)^3/(1 - x*A(x)^7) + x^4*A(x)^4/(1 - x*A(x)^9) + ...
also
B(x) = 1/(1-x*A(x)^2) + x/(1 - x*A(x)^3) + x^2/(1 - x*A(x)^4) + x^3/(1 - x*A(x)^5) + x^4/(1 - x*A(x)^6) + x^5/(1 - x*A(x)^7) + ...
where
B(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 93*x^4 + 602*x^5 + 4406*x^6 + 34666*x^7 + 286098*x^8 + 2443548*x^9 + 21419500*x^10 + 191631430*x^11 + ...
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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+2)) ) - sum(m=0, #A, x^m*Ser(A)^m/(1 - x*Ser(A)^(2*m+1)) ), #A); 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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