The On-Line Encyclopedia of Integer Sequences (OEIS)

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A273925

G.f. satisfies: A( A(x)^2 - A(x)^3 ) = x^2, where A(x) = Sum_{n>=1} a(n)*x^n / 2^A273926(n).
(history; published version)

#23 by Paul D. Hanna at Sun Jun 05 19:05:37 EDT 2016

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#22 by Paul D. Hanna at Sun Jun 05 19:05:32 EDT 2016

FORMULA

(1) A( +sqrt( A( x^2 - x^3 ) ) ) = x.

(2) A( -sqrt( A( x^2 - x^3 ) ) ) = (1 - x - sqrt(1 + 2*x - 3*x^2))/2.

EXAMPLE

such that A( A(x)^2 - A(x)^3 ) = x^2 and A( +sqrt( A(x^2 - x^3) ) ) = x.

A( -sqrt( A(x)^2 - x^3 ) ) ) = - A(x)^2 ) = - + x^2 + x^4 - x^6 3 + 2*x^8 4 - 4*x^10 5 + 9*x^12 6 - 21*x^14 7 + 51*x^16 8 - 127*x^18 9 + 323*x^20 10 - 835*x^22 11 +...+ (-1)^n*A001006(n-2)*x^(2*n) +...

which equals (1 - x^2 - sqrt(1 + 2*x^2 - 3*x^42))/2.

Also, we have

A( A(x)^3 - A(x)^2 ) = (1 - x^2 - sqrt(1 + 2*x^2 - 3*x^4))/2.

Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then B(x) = sqrt( A(x^2 - x^3) ) and begins

then B(x) = sqrt( A(x^2 - x^3) ) and begins

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#21 by Paul D. Hanna at Sun Jun 05 18:59:46 EDT 2016

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#20 by Paul D. Hanna at Sun Jun 05 18:59:41 EDT 2016

FORMULA

(1) A( +sqrt( A( x^2 - x^3 ) ) ) = x.

(2) A( -sqrt( A( x)^3 2 - A(x)^2 3 ) ) ) = (1 - x^2 - sqrt(1 + 2*x^2 - 3*x^4))/2, an integer power series in x^))/2 with Motzkin numbers (A001006) for coefficients.

(3) A( A(x)^3 - A(x)^2 ) = (1 - x^2 - sqrt(1 + 2*x^2 - 3*x^4))/2.

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#19 by Paul D. Hanna at Sun Jun 05 09:11:34 EDT 2016

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#18 by Paul D. Hanna at Sun Jun 05 09:11:30 EDT 2016

FORMULA

(2) A( A(x)^3 - A(x)^2 ) = (1 - x^2 - sqrt(1 + 2*x^2 - 3*x^4))/2 = -x^2 + x^4*M(-x^2) where M(x) = 1 + x*M(x) + x^2*M(, an integer power series in x)^2 is a g.f. of the with Motzkin numbers (A001006) for coefficients.

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#17 by Paul D. Hanna at Sat Jun 04 21:25:48 EDT 2016

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#16 by Paul D. Hanna at Sat Jun 04 21:25:42 EDT 2016

FORMULA

(2) A( A(x)^3 - A(x)^2 ) = (1 - x^2 - sqrt(1 + 2*x^2 - 3*x^4))/2 = -x^2 + x^4*M(-x^2) where M(x) = x 1 + x*M(x) + x^2*M(x)^2 is a g.f. of the Motzkin numbers (A001006).

EXAMPLE

A( A(x)^3 - A(x)^2 ) = -x^2 + x^4 - x^6 + 2*x^8 - 4*x^10 + 9*x^12 - 21*x^14 + 51*x^16 - 127*x^18 + 323*x^20 - 835*x^22 +...+ (-1)^n*A001006(n-2)*x^(2*n) +...

which equals (1 - x^2 - sqrt(1 + 2*x^2 - 3*x^4))/2.

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#15 by Paul D. Hanna at Sat Jun 04 18:45:20 EDT 2016

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#14 by Paul D. Hanna at Sat Jun 04 18:44:48 EDT 2016

FORMULA

G.f. A(x) = Sum_{n>=1} a(n)*x^n / 2^A273926(n) satisfies:

(1) A( sqrt( A( x^2 - x^3 ) ) ) = x, where A(x) = Sum_{n>=1} a(n)*x^n / 2^A273926(n).

(2) A( A(x)^3 - A(x)^2 ) = -M(-x^2) where M(x) = x + x*M(x) + x^2*M(x)^2 is a g.f. of the Motzkin numbers (A001006).

EXAMPLE

The g.f. is related to the Motzkin numbers by the relation:

A( A(x)^3 - A(x)^2 ) = -x^2 + x^4 - x^6 + 2*x^8 - 4*x^10 + 9*x^12 - 21*x^14 + 51*x^16 - 127*x^18 + 323*x^20 - 835*x^22 +...+ (-1)^n*A001006(n)*x^(2*n) +...

A relevant series begins:

The g.f. is related to the Motzkin numbers by the series:

A( A(x)^3 - A(x)^2 ) = -x^2 + x^4 - x^6 + 2*x^8 - 4*x^10 + 9*x^12 - 21*x^14 + 51*x^16 - 127*x^18 + 323*x^20 - 835*x^22 +...+ (-1)^n*A001006(n)*x^(2*n) +...

CROSSREFS

Cf. A273926, A001006.

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