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A275756
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G.f. satisfies: A(x) = x + A( A(x)^2 - A(x)^6 ).
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4
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1, 1, 2, 6, 20, 71, 264, 1018, 4032, 16305, 67042, 279444, 1178088, 5014596, 21521488, 93027025, 404630318, 1769704106, 7778030834, 34335337802, 152168657438, 676796514510, 3019945599904, 13515300673984, 60649985907334, 272847379282493, 1230295797205452, 5559373120441048, 25171114275512520, 114177375142080814, 518806321789317040, 2361183952087172306, 10762422470020855820, 49125407360603361370, 224533932290057629076, 1027553322543206612019, 4708070541211739962738, 21595828228486254332762
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OFFSET
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1,3
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COMMENTS
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Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
Compare to: C(x) = x + C( C(x)^3 - C(x)^9 ) holds when C(x) = x + C(x)^3 is a g.f. of the ternary tree numbers (A001764).
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LINKS
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FORMULA
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G.f. satisfies:
(1) A(x - A(x^2 - x^6)) = x.
(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^2-x^6)^(n+1) / (n+1)!.
(3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^2-x^6)^(n+1)/x / (n+1)! ).
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EXAMPLE
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G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 71*x^6 + 264*x^7 + 1018*x^8 + 4032*x^9 + 16305*x^10 + 67042*x^11 + 279444*x^12 + 1178088*x^13 + 5014596*x^14 +...
such that A(x) = x + A( A(x)^2 - A(x)^6 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 56*x^6 + 206*x^7 + 786*x^8 + 3088*x^9 + 12408*x^10 + 50754*x^11 + 210639*x^12 + 884784*x^13 + 3754424*x^14 +...
A(x)^6 = x^6 + 6*x^7 + 27*x^8 + 116*x^9 + 495*x^10 + 2112*x^11 + 9035*x^12 + 38820*x^13 + 167628*x^14 + 727480*x^15 + 3172455*x^16 +...
A(x^2 - x^6) = x^2 + x^4 + x^6 + 4*x^8 + 14*x^10 + 48*x^12 + 170*x^14 + 628*x^16 + 2382*x^18 + 9202*x^20 + 36098*x^22 + 143484*x^24 + 576638*x^26 + 2339050*x^28 +...
where Series_Reversion(A(x)) = x - A(x^2 - x^6).
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PROG
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(PARI) {a(n) = my(A=x); for(i=1, n, A = x + subst(A, x, A^2 - A^6 +x*O(x^n))); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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