A295762
G.f. A(x) satisfies: A(x - 2*A(x^2)) = x + A(x^2).
Original entry on oeis.org
1, 3, 12, 69, 444, 3060, 22104, 165195, 1266636, 9908196, 78760920, 634379124, 5166150000, 42465716328, 351876854448, 2936058188877, 24648274487556, 208040487845076, 1764376309044792, 15027939263874132, 128495423551583664, 1102547377746843624, 9490542912076091184, 81931260285359287812, 709199467337528862768, 6153967855892699398368, 53521531522907694320928, 466461452477641527148344
Offset: 1
G.f.: A(x) = x + 3*x^2 + 12*x^3 + 69*x^4 + 444*x^5 + 3060*x^6 + 22104*x^7 + 165195*x^8 + 1266636*x^9 + 9908196*x^10 + 78760920*x^11 + 634379124*x^12 +...
such that A(x - 2*A(x^2)) = x + A(x^2).
RELATED SERIES.
A(x - 2*A(x^2)) = x + x^2 + 3*x^4 + 12*x^6 + 69*x^8 + 444*x^10 + 3060*x^12 + 22104*x^14 + 165195*x^16 + 1266636*x^18 + 9908196*x^20 +...
which equals x + A(x^2).
Series_Reversion( x - 2*A(x^2) ) = x + 2*x^2 + 8*x^3 + 46*x^4 + 296*x^5 + 2040*x^6 + 14736*x^7 + 110130*x^8 + 844424*x^9 + 6605464*x^10 + 52507280*x^11 + 422919416*x^12 +...
which equals (2*A(x) + x)/3.
A( (x + 2*A(x))^2/9 ) = x^2 + 4*x^3 + 23*x^4 + 148*x^5 + 1020*x^6 + 7368*x^7 + 55065*x^8 + 422212*x^9 + 3302732*x^10 + 26253640*x^11 + 211459708*x^12 +...
which equals (A(x) - x)/3.
Odd terms seem to occur only at positions 2^n, n>=0, beginning:
[1, 3, 69, 165195, 2936058188877, 2740954751925406954539018771, 6899036855844990995854505818787102393537422152828959745477, ...].
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nmax = 28; sol = {a[1] -> 1}; Do[A[x_] = Sum[a[k] x^k, {k, 1, n}] /. sol; eq = CoefficientList[A[x - 2 A[x^2]] - (x + A[x^2]) + O[x]^(n+1) // Normal, x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. Rule -> Set;
a /@ Range[nmax] (* Jean-François Alcover, Nov 03 2019 *)
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{a(n) = my(A=x); for(i=1,n, A = -x/2 + 3/2*serreverse(x - 2*subst(A,x,x^2) +x^2*O(x^n)) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
A177408
G.f. satisfies: A(x) = x + A( 4*A(x)^4 )^(1/2).
Original entry on oeis.org
1, 2, 8, 40, 224, 1352, 8576, 56352, 380160, 2617584, 18320384, 129950912, 932114432, 6749344832, 49268899840, 362189529344, 2678989406208, 19923485019840, 148887398711296, 1117452514604800, 8419605676818432
Offset: 1
G.f.: A(x) = x + 2*x^2 + 8*x^3 + 40*x^4 + 224*x^5 + 1352*x^6 +...
Related expansions:
. A(4A(x)^4) = 4*x^4 + 32*x^5 + 224*x^6 + 1536*x^7 + 10592*x^8 +...
. A(x)^4 = x^4 + 8*x^5 + 56*x^6 + 384*x^7 + 2640*x^8 + 18336*x^9 +...
. A(4x^4)^(1/2) = 2*x^2 + 8*x^6 + 112*x^10 + 2112*x^14 + 45760*x^18 +...
...
The series reversion is defined by R(x) = x - A(4x^4)^(1/2) where:
. R(x) = x - 2*x^2 - 8*x^6 - 112*x^10 - 2112*x^14 - 45760*x^18 -...
. x/R(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4 + 40*x^5 + 96*x^6 + 224*x^7 +...
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{a(n)=local(A=x+x^2);for(i=1,n,A=x+subst(A,x,4*(A+x*O(x^n))^4)^(1/2));polcoeff(A,n)}
A378255
G.f. A(x) satisfies A(x) = x + A(A(x)^2) + A(A(x)^3).
Original entry on oeis.org
1, 1, 3, 11, 44, 193, 882, 4178, 20305, 100694, 507493, 2591897, 13384911, 69773480, 366661360, 1940336952, 10331179153, 55306072496, 297499878304, 1607212566176, 8716586959731, 47440220540922, 259021671704538, 1418386679682870, 7787843448380598, 42865830110488341, 236480195092162079
Offset: 1
G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 193*x^6 + 882*x^7 + 4178*x^8 + 20305*x^9 + 100694*x^10 + 507493*x^11 + 2591897*x^12 + ...
where A(x) = x + A(A(x)^2) + A(A(x)^3).
RELATED SERIES.
A(A(x)^2) = x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 140*x^6 + 642*x^7 + 3044*x^8 + 14810*x^9 + 73508*x^10 + 370744*x^11 + 1894641*x^12 + ...
A(A(x)^3) = x^3 + 3*x^4 + 12*x^5 + 53*x^6 + 240*x^7 + 1134*x^8 + 5495*x^9 + 27186*x^10 + 136749*x^11 + 697256*x^12 + ...
A(x^2) + A(x^3) = x^2 + x^3 + x^4 + 4*x^6 + 11*x^8 + 3*x^9 + 44*x^10 + 204*x^12 + 882*x^14 + 44*x^15 + 4178*x^16 + 20498*x^18 + 100694*x^20 + 882*x^21 + 507493*x^22 + 2596075*x^24 + ...
where A(x - A(x^2) - A(x^3)) = x.
SPECIFIC VALUES.
A(t) = 2/7 at t = 0.17113633000646334369180481612349578624783801172280...
where t = 2/7 - A(4/49) - A(8/343).
A(t) = 1/4 at t = 0.16675404370583413662402450218565513714831350182372...
where t = 1/4 - A(1/16) - A(1/64).
A(t) = 1/5 at t = 0.15010878278748588281213370320635881316994691322045...
where t = 1/5 - A(1/25) - A(1/125).
A(t) = 1/6 at t = 0.13339423917640795169936128266096825782919420931151...
where t = 1/6 - A(1/36) - A(1/216).
A(t) = 1/7 at t = 0.11908088070152938397080640031639630333627904913531...
A(t) = 1/8 at t = 0.10716175341844903845196692277233592851173749413122...
A(t) = 1/9 at t = 0.09723347082793633309633745814458884367513391389109...
A(t) = 1/10 at t = 0.0888958823866966406913909794974554201549795441139...
A(1/16) = 0.0673646719229547504984252010615186334929817484538318...
A(1/25) = 0.0418256346622057039599677785843885345768765252605719...
A(1/36) = 0.0286210615541792671675075085528989453238496370093068...
A(1/64) = 0.0158812843712111128775502967528262293587047497224419...
A(1/125) = 0.0080655825503084132278985182092526522531765615189742...
A(1/216) = 0.0046513659360794477997978754527994635136228203458472...
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\\ Formula: A(x) = Series_Reversion(x - A(x^2) - A(x^3)) \\
{a(n) = my(A=x+x^2); for(i=0, n, A = serreverse(x - subst(A, x, x^2 +x^2*O(x^n)) - subst(A, x, x^3 +x^2*O(x^n)))) ; polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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\\ Formula: A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (A(x^2) + A(x^3))^n/n! \\
{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2+x*O(x^n)); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, (subst(A, x, x^2+x*O(x^n)) + subst(A, x, x^3+x*O(x^n)))^m)/m!)+x*O(x^n)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
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