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%I #7 Nov 23 2017 06:14:59
%S 1,1,1,4,10,32,95,306,978,3235,10767,36470,124514,429648,1492944,
%T 5225700,18396350,65115694,231555165,826956617,2964543205,10664540170,
%U 38484972969,139281469165,505408580484,1838442927937,6702466323520,24486411113076,89630823136513,328680670354328,1207323483992684,4441801238353311,16365832987077134,60384021404260146,223087697417538491
%N G.f. A(x) satisfies: A(x) = A(x)^2 - x*A(x)^3 + x^2.
%H Paul D. Hanna, <a href="/A295404/b295404.txt">Table of n, a(n) for n = 0..500</a>
%F G.f. A(x) satisfies: A(x) = 1 + x*A(x)^2 - x^2/A(x).
%F a(n) ~ sqrt((s^3 - 2*r)/(Pi*(3*r*s - 1))) / (2*n^(3/2)*r^(n - 1/2)), where r = 0.2590976379022320530812109572925567785373263490686... and s = 1.89364715749587181948481325332597309754099061462... are real roots of the system of equations r^2 + s^2 = s + r*s^3, 1 + 3*r*s^2 = 2*s. - _Vaclav Kotesovec_, Nov 23 2017
%e G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 10*x^4 + 32*x^5 + 95*x^6 + 306*x^7 + 978*x^8 + 3235*x^9 + 10767*x^10 + 36470*x^11 + 124514*x^12 + 429648*x^13 + 1492944*x^14 + 5225700*x^15 + 18396350*x^16 + 65115694*x^17 + 231555165*x^18 + 826956617*x^19 + 2964543205*x^20 +...
%e such that A(x) = A(x)^2 - x*A(x)^3 + x^2.
%e RELATED SERIES.
%e 1/A(x) = 1 - x - 3*x^3 - 3*x^4 - 16*x^5 - 32*x^6 - 121*x^7 - 329*x^8 - 1138*x^9 - 3546*x^10 - 12097*x^11 - 40112*x^12 +...
%e A(x)^2 = 1 + 2*x + 3*x^2 + 10*x^3 + 29*x^4 + 92*x^5 + 290*x^6 + 946*x^7 + 3114*x^8 + 10438*x^9 + 35332*x^10 + 120968*x^11 + 417551*x^12 +...
%e A(x)^3 = 1 + 3*x + 6*x^2 + 19*x^3 + 60*x^4 + 195*x^5 + 640*x^6 + 2136*x^7 + 7203*x^8 + 24565*x^9 + 84498*x^10 + 293037*x^11 + 1023184*x^12 +...
%e where A(x) = 1 + x*A(x)^2 - x^2/A(x).
%e Series_Reversion(x*A(x)) = x - x^2 + x^3 - 4*x^4 + 10*x^5 - 32*x^6 + 107*x^7 - 360*x^8 + 1270*x^9 - 4544*x^10 + 16537*x^11 - 61092*x^12 + 228084*x^13 - 860056*x^14 + 3269994*x^15 +...+ (-1)^(n-1)*A137954(n-1)*x^n +...
%o (PARI) {a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A^2 - x^2/A +x*O(x^n)); polcoeff(G=A, n)}
%o for(n=0,40,print1(a(n),", "))
%Y Cf. A137954.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Nov 21 2017
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