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A340329
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G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)).
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6
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1, 2, 4, 11, 33, 111, 395, 1461, 5565, 21654, 85694, 343819, 1395221, 5716363, 23613391, 98238069, 411241397, 1730976298, 7321464729, 31102459844, 132645440213, 567712900716, 2437610016301, 10497289161264, 45327443941706, 196210780501951, 851293469453768
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OFFSET
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0,2
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COMMENTS
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The g.f. of this sequence is motivated by the following identity:
Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;
here, p = x, q = x*A(x), and r = x.
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 0..500
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FORMULA
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G.f. A(x) satisfies the following relations.
(1) A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)).
(2) A(x) = Sum_{n>=0} x^n*A(x)^n / (1 - x^(n+1)).
(3) A(x) = Sum_{n>=0} x(n^2+2*n) * A(x)^n * (1 - x^(2*n+2)*A(x)) / ((1 - x^(n+1))*(1 - x^(n+1)*A(x))). - Paul D. Hanna, Feb 20 2021
a(n) ~ c * d^n / n^(3/2), where d = 4.5952034330689668887505654340441530851246660208... and c = 0.7049561865662193122992597796630536353257... - Vaclav Kotesovec, Jan 07 2021
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EXAMPLE
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G.f.: A(x) = 1 + 2*x + 4*x^2 + 11*x^3 + 33*x^4 + 111*x^5 + 395*x^6 + 1461*x^7 + 5565*x^8 + 21654*x^9 + 85694*x^10 + ...
where
A(x) = 1/(1 - x*A(x)) + x/(1 - x^2*A(x)) + x^2/(1 - x^3*A(x)) + x^3/(1 - x^4*A(x)) + x^4/(1 - x^5*A(x)) + ...
also
A(x) = 1/(1 - x) + x*A(x)/(1 - x^2) + x^2*A(x)^2/(1 - x^3) + x^3*A(x)^3/(1 - x^4) + x^4*A(x)^4/(1 - x^5) + ...
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PROG
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(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^m / (1 - x^(m+1)*A +x*O(x^n)) )); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, x^m*A^m / (1 - x^(m+1) +x*O(x^n)) )); ; polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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Cf. A340338, A340355,
Sequence in context: A220902 A249945 A208799 * A268323 A268324 A268325
Adjacent sequences: A340326 A340327 A340328 * A340330 A340331 A340332
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Jan 05 2021
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STATUS
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approved
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