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A340329 G.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n / (1 - x^(n+1)*A(x)). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

FORMULA

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

EXAMPLE

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) + ...

PROG

(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), ", "))

CROSSREFS

Cf. A340338, A340355,

Sequence in context: A220902 A249945 A208799 * A268323 A268324 A268325

Adjacent sequences:  A340326 A340327 A340328 * A340330 A340331 A340332

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 05 2021

STATUS

approved

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Last modified April 15 06:12 EDT 2021. Contains 342975 sequences. (Running on oeis4.)