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A356782
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Coefficients in function A(x) such that A(x) = x * Product_{n>=0} (1 + 2*A(x)^(2^n)).
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4
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1, 2, 6, 24, 106, 496, 2428, 12288, 63762, 337392, 1813628, 9876096, 54365876, 302037408, 1691327224, 9536234496, 54093070626, 308474110000, 1767481876540, 10170367611008, 58746459504884, 340513035730944, 1979964903739992, 11546094361266176, 67509252360531940
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OFFSET
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1,2
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A(x)^2 = A( x*A(x) + 2*x*A(x)^2 ).
(2) A(x)^4 = A( x*A(x)^3*(1 + 2*A(x))*(1 + 2*A(x)^2) ).
(3) A(x) = x * Product_{n>=0} (1 + 2*A(x)^(2^n)).
(4) A(x) = x * Sum_{n>=0} 2^A000120(n) * A(x)^n, where A000120(n) = number of 1's in binary expansion of n.
(5) A(x) = x * Sum_{n>=0} A001316(n) * A(x)^n, where A001316(n) = number of odd entries in row n of Pascal's triangle.
(6) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 2*x^(2^n)) ).
a(n) ~ c * d^n / n^(3/2), where d = 6.21914802593104186425760427502624741814921... and c = 0.11969150889542130656096151211011193088986347... - Vaclav Kotesovec, Apr 05 2024
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EXAMPLE
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G.f.: A(x) = x + 2*x^2 + 6*x^3 + 24*x^4 + 106*x^5 + 496*x^6 + 2428*x^7 + 12288*x^8 + 63762*x^9 + 337392*x^10 + 1813628*x^11 + 9876096*x^12 + ...
such that
A(x) = x * (1 + 2*A(x)) * (1 + 2*A(x)^2) * (1 + 2*A(x)^4) * (1 + 2*A(x)^8) * (1 + 2*A(x)^16) * ... * (1 + 2*A(x)^(2^n)) * ...
which is equivalent to
A(x) = x * (1 + 2*A(x) + 2*A(x)^2 + 4*A(x)^3 + 2*A(x)^4 + 4*A(x)^5 + 4*A(x)^6 + 8*A(x)^7 + 2*A(x)^8 + 4*A(x)^9 + 4*A(x)^10 + 8*A(x)^11 + 4*A(x)^12 + 8*A(x)^13 + 8*A(x)^14 + 16*A(x)^15 + ... + 2^A000120(n)*A(x)^n + ...).
Also, g.f. A(x) satisfies
A(x)^2 = A( x*A(x) + 2*x*A(x)^2 )
where
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 72*x^5 + 344*x^6 + 1704*x^7 + 8688*x^8 + 45328*x^9 + 240856*x^10 + 1298984*x^11 + 7092544*x^12 + ...
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PROG
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(PARI) {a(n) = my(A = serreverse( x/prod(k=0, #binary(n), 1 + 2*x^(2^k) +x*O(x^n)) ));
polcoeff(A, n)}
for(n=1, 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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