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A291743 G.f. A(x) satisfies: A(x - 2*x*A(x)) = x + x*A(x). 7
1, 3, 21, 213, 2649, 37683, 591345, 10018881, 180728985, 3438000843, 68495707461, 1421921135541, 30636666967857, 683015673332955, 15717420795468609, 372590181990474273, 9083851303291071633, 227458241320778287731, 5842792897560424283157, 153812115285865111498869, 4146016380695723021134761, 114341728157990237568733731, 3224107003868451067670682993 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..200

FORMULA

G.f. A(x) satisfies:

(1) A(x) = 3/2 * Series_Reversion( x - 2*x*A(x) ) - x/2.

(2) A(x) = x * (1 + A(B(x))) / (1 - 2*A(B(x))), where B(x) = (x + 2*A(x))/3.

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

EXAMPLE

G.f.: A(x) = x + 3*x^2 + 21*x^3 + 213*x^4 + 2649*x^5 + 37683*x^6 + 591345*x^7 + 10018881*x^8 + 180728985*x^9 + 3438000843*x^10 + 68495707461*x^11 + 1421921135541*x^12 +...

such that A(x - 2*x*A(x)) = x + x*A(x).

RELATED SERIES.

A(x - 2*x*A(x)) = x + x^2 + 3*x^3 + 21*x^4 + 213*x^5 + 2649*x^6 + 37683*x^7 + 591345*x^8 +...

which equals x + x*A(x).

Series_Reversion( x - x*A(x) ) = x + 2*x^2 + 14*x^3 + 142*x^4 + 1766*x^5 + 25122*x^6 + 394230*x^7 + 6679254*x^8 + 120485990*x^9 + 2292000562*x^10 + 45663804974*x^11 + 947947423694*x^12 +...

which equals (x + 2*A(x))/3.

A( (x + 2*A(x))/3 ) = x + 5*x^2 + 47*x^3 + 577*x^4 + 8273*x^5 + 131865*x^6 + 2275299*x^7 + 41821401*x^8 + 810261881*x^9 + 16426252597*x^10 + 346605967151*x^11 + 7582215510713*x^12 +...

which equals (A(x) - x) / (2*A(x) + x).

PROG

(PARI) {a(n) = my(A=x); for(i=1, n, A = 3/2*serreverse( x - 2*x*A +x*O(x^n) ) - x/2 ); polcoeff(A, n)}

for(n=1, 30, print1(a(n), ", "))

(PARI) {a(n) = my(A=x, B); for(i=1, n, B = (x + 2*A)/3 +x*O(x^n); A = x*(1 + subst(A, x, B))/(1 - 2*subst(A, x, B)) ); polcoeff(A, n)}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A291744, A276358.

Sequence in context: A123691 A087918 A088926 * A120972 A168479 A158838

Adjacent sequences:  A291740 A291741 A291742 * A291744 A291745 A291746

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 30 2017

STATUS

approved

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Last modified February 23 00:28 EST 2020. Contains 332157 sequences. (Running on oeis4.)