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
Offset: 0
Keywords
Examples
A(x) = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 412*x^5 + 4120*x^6 +... The g.f. of A120955 is: x/series_reversion(x*A(x)) = 1 + x + x^2 + 4*x^3 + 25*x^4 + 206*x^5 +... Compare terms to see that A120955(n) = a(n)/2 for n>=2. A(x*A(x)) = 1 + x + 3*x^2 + 14*x^3 + 92*x^4 + 774*x^5 +... A(x)*(2-x) = 2 + x + 3*x^2 + 14*x^3 + 92*x^4 + 774*x^5 +... Contribution from _Paul D. Hanna_, Sep 04 2010: (Start) Let G(x) = x*A(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)
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..400
Crossrefs
Cf. A120955.
Programs
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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]} for(n=0,30, print1(a(n),", ")) -
PARI
/* Prints N terms using x/Series_Reversion(x*A(x)) = (A(x) + 1+x)/2 */ N = 30; {A=[1,1]; for(i=1,N, A = concat(A, -2*Vec(x/serreverse(x*Ser(concat(A,0))))[#A+1]); print1(i,",") );A} \\ Paul D. Hanna, Sep 21 2019
Formula
a(n) = 2*A120955(n) for n>=2.
G.f. A(x) satisfies:
(1) A( 2x/(A(x) + 1+x) ) = (A(x) + 1+x)/2.
(2) A(x) = F(x*A(x)) and F(x) = A(x/F(x)) where F(x) = g.f. of A120955.
(3) A(x) = (1 + A(x*A(x))) / (2-x).
(4) A(x) = 1 + Sum_{n>=0} G_n(x)/2^(n+1) where G(x)=x*A(x) and G_{n+1}(x) = G_n(x*A(x)) denotes iteration with G_0(x)=x. [From Paul D. Hanna, Sep 04 2010]
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