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A273030 G.f. A(x) satisfies: A(x)^3 = x * A( (A(x) + A(x)^2)^2 ). 0

%I

%S 1,2,7,34,189,1132,7134,46642,313468,2152318,15032964,106474940,

%T 762949599,5520867592,40287276592,296133267074,2190621494149,

%U 16296005417256,121830586282884,914879338223438,6897776776248601,52194621068657552,396250628678762793,3017289164647604068,23038674915285329005,176358202113609998074,1353151958449050620098,10404826279377895563688,80166547565004475870252,618815710970799456428120

%N G.f. A(x) satisfies: A(x)^3 = x * A( (A(x) + A(x)^2)^2 ).

%F G.f. A(x) satisfies: A( x^3 / A(x^2 + 2*x^3 + x^4) ) = x.

%e G..f.: A(x) = x + 2*x^2 + 7*x^3 + 34*x^4 + 189*x^5 + 1132*x^6 + 7134*x^7 + 46642*x^8 + 313468*x^9 + 2152318*x^10 + 15032964*x^11 + 106474940*x^12 +...

%e such that A(x)^3 = x * A( (A(x) + A(x)^2)^2 ).

%e RELATED SERIES.

%e A(x)^2 = x^2 + 4*x^3 + 18*x^4 + 96*x^5 + 563*x^6 + 3496*x^7 + 22598*x^8 + 150520*x^9 + 1026077*x^10 + 7124504*x^11 + 50213484*x^12 + 358312064*x^13 +...

%e A(x)^3 = x^3 + 6*x^4 + 33*x^5 + 194*x^6 + 1206*x^7 + 7794*x^8 + 51859*x^9 + 353028*x^10 + 2447694*x^11 + 17227300*x^12 + 122769939*x^13 + 884167752*x^14 +...

%e A( (A(x) + A(x)^2)^2 ) = x^2 + 6*x^3 + 33*x^4 + 194*x^5 + 1206*x^6 + 7794*x^7 + 51859*x^8 + 353028*x^9 + 2447694*x^10 + 17227300*x^11 + 122769939*x^12 +...

%e (A(x) + A(x)^2)^2 = x^2 + 6*x^3 + 31*x^4 + 170*x^5 + 1003*x^6 + 6244*x^7 + 40404*x^8 + 269190*x^9 + 1834781*x^10 + 12735668*x^11 + 89726127*x^12 +...

%e The square-root of x*A(x) is an integer series:

%e sqrt( x*A(x) ) = x + x^2 + 3*x^3 + 14*x^4 + 76*x^5 + 448*x^6 + 2793*x^7 + 18120*x^8 + 121075*x^9 + 827574*x^10 + 5759383*x^11 + 40671931*x^12 + 290718799*x^13 +...

%e A((x + x^2)^2) = x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 19*x^6 + 50*x^7 + 141*x^8 + 412*x^9 + 1246*x^10 + 3836*x^11 + 12024*x^12 + 38168*x^13 + 122488*x^14 +...

%e sqrt( A((x + x^2)^2) ) = x + x^2 + x^3 + 3*x^4 + 6*x^5 + 16*x^6 + 44*x^7 + 128*x^8 + 385*x^9 + 1177*x^10 + 3674*x^11 + 11606*x^12 + 37107*x^13 + 119819*x^14 +...

%e Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then

%e B(x) = x - 2*x^2 + x^3 - 4*x^4 + 2*x^5 - 12*x^6 - 10*x^7 - 64*x^8 - 147*x^9 - 498*x^10 - 1493*x^11 - 4732*x^12 - 15050*x^13 - 48436*x^14 - 157400*x^15 +...

%e where B(x) = x^3 / A((x + x^2)^2),

%e also, B(x^3/B(x)) = (x + x^2)^2.

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

%o for(n=1, 40, print1(a(n), ", "))

%Y Cf. A272485.

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 15 2016

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Last modified March 4 01:48 EST 2021. Contains 341773 sequences. (Running on oeis4.)