A120956 - OEIS

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A120956

G.f. A(x) satisfies x/series_reversion(x*A(x)) = (A(x) + 1+x)/2.

1, 1, 2, 8, 50, 412, 4120, 47840, 628130, 9164600, 146786980, 2557718352, 48147082520, 973612557504, 21050077835440, 484637221115520, 11839623684281890, 305949448095405252, 8339153054042801704 (list; graph; refs; listen; history; internal format)

	OFFSET	`0,3`
	COMMENTS	`The g.f. for A120955 = x/series_reversion(x*A(x)) = (A(x) + 1+x)/2.`
	FORMULA	`a(n) = 2A120955(n) for n>=2. G.f. A(x) satisfies: A( 2x/(A(x) + 1+x) ) = (A(x) + 1+x)/2. G.f. A(x) satisfies: A(x) = F(xA(x)) and F(x) = A(x/F(x)) where F(x) = g.f. of A120955.` `G.f. satisfies: A(x) = (1 + A(xA(x)) )/(2-x).` `G.f. satisfies: A(x) = 1 + Sum_{n>=0} G_n(x)/2^(n+1) where G(x)=xA(x) and G_{n+1}(x) = G_n(x*A(x)) denotes iteration with G_0(x)=x. [From Paul D. Hanna (pauldhanna(AT)juno.com), Sep 04 2010]`
	EXAMPLE	`A(x) = 1 + x + 2x^2 + 8x^3 + 50x^4 + 412x^5 + 4120x^6 +...` `The g.f. of A120955 is:` `x/series_reversion(xA(x)) = 1 + x + x^2 + 4x^3 + 25x^4 + 206x^5 +...` `Compare terms to see that A120955(n) = a(n)/2 for n>=2.` `A(xA(x)) = 1 + x + 3x^2 + 14x^3 + 92x^4 + 774x^5 +...` `A(x)(2-x) = 2 + x + 3x^2 + 14x^3 + 92x^4 + 774x^5 +...` `Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Sep 04 2010: (Start)` `Let G(x) = xA(x), then` `A(x) = 1 + G(x)/2 + G(G(x))/2^2 + G(G(G(x)))/2^3 + G(G(G(G(x))))/2^4 + G(G(G(G(G(x)))))/2^5 +...` `The table of coefficients in the iterations of G(x)=x*A(x) begin:` `[1, 1, 2, 8, 50, 412, 4120, 47840, 628130, ...];` `[1, 2, 6, 27, 170, 1380, 13580, 155568, 2020526, ...];` `[1, 3, 12, 63, 422, 3482, 34208, 389007, 5010678, ...];` `[1, 4, 20, 122, 892, 7690, 76900, 878032, 11284106, ...];` `[1, 5, 30, 210, 1690, 15490, 160464, 1864844, 24130948, ...];` `[1, 6, 42, 333, 2950, 29002, 315184, 3775392, 49699640, ...];` `[1, 7, 56, 497, 4830, 51100, 587104, 7318983, 98962072, ...];` `[1, 8, 72, 708, 7512, 85532, 1043032, 13621120, 190640924, ...];` `[1, 9, 90, 972, 11202, 137040, 1776264, 24394608, 355390206, ...]; ...` `in which the following sum along column k equals a(k+1):` `a(2) = 2 = 1/2 + 2/4 + 3/8+ 4/16 + 5/32 + 6/64 +...` `a(3) = 8 = 2/2 + 6/4 + 12/8 + 20/16 + 30/32 + 42/64 + ...` `a(4) = 50 = 8/2 + 27/4 + 63/8 + 122/16 + 210/32 + 333/64 +...` `a(5) = 412 = 50/2 + 170/4 + 422/8 + 892/16 + 1690/32 + 2950/64 +... (End)`
	PROG	`(PARI) {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, t); A[ #A]=subst(Vec(serreverse(x/Ser(A)))[ #A], t, 0)); Vec(serreverse(x/Ser(A)))[n+1]}`
	CROSSREFS	`Cf. A120955.` `Sequence in context: A034491 A186182 A121677 * A000557 A193352 A002801` `Adjacent sequences: A120953 A120954 A120955 * A120957 A120958 A120959`
	KEYWORD	`nonn`
	AUTHOR	`Paul D. Hanna (pauldhanna(AT)juno.com), Jul 19 2006`

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Last modified February 14 03:37 EST 2012. Contains 205570 sequences.