This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
G.f.: A(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +... where A(x) = x + A(A(x))^2: A(A(x)) = x + 2*x^2 + 10*x^3 + 69*x^4 + 568*x^5 + 5250*x^6 + 52792*x^7 +... A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +... The g.f. satisfies the series: A(x) = x + A(x)^2 + d/dx A(x)^4/2! + d^2/dx^2 A(x)^6/3! + d^3/dx^3 A(x)^8/4! +... Logarithmic series: log(A(x)/x) = A(x)^2/x + [d/dx A(x)^4/x]/2! + [d^2/dx^2 A(x)^6/x]/3! + [d^3/dx^3 A(x)^8/x]/4! +... Also, A(x) = x*G(A(x)^2/x) where G(x) = x/A(x/G(x)^2) is the g.f. of A212411: G(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 + 15261*x^7 +... Also, A(x)^2 = x*F(A(x)) where F(x) is the g.f. of A213628: F(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 616*x^6 + 5072*x^7 + 46013*x^8 +...
terms = 22; A[] = 0; Do[A[x] = x + A[A[x]]^2 + O[x]^(terms+1) // Normal, terms+1]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Jan 09 2018 *)
{a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^2+x*O(x^n))); polcoeff(A, n))}
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+sum(m=1, n, Dx(m-1, A^(2*m))/m!)+x*O(x^n)); polcoeff(A, n)}
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x*exp(sum(m=1, n, Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1,21,print1(a(n),", "))
b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 2*j))); a(n) = b(n-1, 1); \\ Seiichi Manyama, Jun 05 2025
G.f.: A(x) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1148*x^5 + 14156*x^6 + 191400*x^7 + 2775930*x^8 + 42585412*x^9 + 684496988*x^10 + 11449962008*x^11 + 198331811356*x^12 +... such that A(x - A(x)^2) = x + A(x)^2. RELATED SERIES. Series_Reversion(x - A(x)^2) = x + x^2 + 6*x^3 + 53*x^4 + 574*x^5 + 7078*x^6 + 95700*x^7 + 1387965*x^8 + 21292706*x^9 + 342248494*x^10 +... which equals (A(x) + x)/2. A( (A(x) + x)/2 ) = x + 3*x^2 + 22*x^3 + 221*x^4 + 2634*x^5 + 35086*x^6 + 506356*x^7 + 7773279*x^8 + 125441594*x^9 + 2110832382*x^10 +... which equals sqrt( (A(x) - x)/2 ). Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x, R(x) = x - 2*x^2 - 4*x^3 - 26*x^4 - 228*x^5 - 2396*x^6 - 28440*x^7 - 369114*x^8 - 5135468*x^9 - 75602108*x^10 - 1167066216*x^11 - 18768202924*x^12 +... then Series_Reversion(x + A(x)^2) = x/2 + R(x)/2.
m = 26; A[_] = 0;
Do[A[x_] = x + 2 A[x/2 + A[x]/2]^2 + O[x]^(m+1) // Normal, {m+1}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
{a(n) = my(A=[1], F=x); for(i=1,n, A=concat(A,0); F=x*Ser(A); A[#A] = -polcoeff(subst(F,x,x-F^2) - F^2,#A) );A[n]}
for(n=1,30,print1(a(n),", "))
G.f.: A(x) = x + 3*x^2 + 24*x^3 + 276*x^4 + 3858*x^5 + 61092*x^6 + 1056816*x^7 + 19550475*x^8 + 381543576*x^9 + 7782820548*x^10 + 164842646424*x^11 + 3607654164924*x^12 +... such that A(x - A(x)^2) = x + 2*A(x)^2. RELATED SERIES. Note that Series_Reversion(x - A(x)^2) = 2*x/3 + A(x)/3, which begins: Series_Reversion(x - A(x)^2) = x + x^2 + 8*x^3 + 92*x^4 + 1286*x^5 + 20364*x^6 + 352272*x^7 + 6516825*x^8 + 127181192*x^9 + 2594273516*x^10 + 54947548808*x^11 + 1202551388308*x^12 +... Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x, R(x) = x - 3*x^2 - 6*x^3 - 51*x^4 - 564*x^5 - 7416*x^6 - 109764*x^7 - 1772028*x^8 - 30603930*x^9 - 558238326*x^10 - 10659285096*x^11 - 211688430204*x^12 +... then Series_Reversion(x + 2*A(x)^2) = x/3 + 2*R(x)/3.
m = 24; A[_] = 0;
Do[A[x_] = x + 3 A[2 x/3 + A[x]/3]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
{a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-F^2) - 2*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
G.f.: A(x) = x + 3*x^2 + 30*x^3 + 447*x^4 + 8202*x^5 + 171846*x^6 + 3956796*x^7 + 97916895*x^8 + 2567551890*x^9 + 70655670690*x^10 + 2026596875268*x^11 + 60282027684678*x^12 +... such that A(x - 2*A(x)^2) = x + A(x)^2. RELATED SERIES. Note that Series_Reversion(x - 2*A(x)^2) = x/3 + 2*A(x)/3, which begins: Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 20*x^3 + 298*x^4 + 5468*x^5 + 114564*x^6 + 2637864*x^7 + 65277930*x^8 + 1711701260*x^9 + 47103780460*x^10 + 1351064583512*x^11 + 40188018456452*x^12 +... Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x, R(x) = x - 3*x^2 - 12*x^3 - 132*x^4 - 1992*x^5 - 36144*x^6 - 742176*x^7 - 16688880*x^8 - 402824928*x^9 - 10300868160*x^10 - 276531035520*x^11 - 7742210941056*x^12 +... then Series_Reversion(x + A(x)^2) = 2*x/3 + R(x)/3.
m = 24; A[_] = 0;
Do[A[x_] = x + 3 A[x/3 + 2 A[x]/3]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
{a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-2*F^2) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
G.f.: A(x) = x + 4*x^2 + 40*x^3 + 564*x^4 + 9592*x^5 + 184008*x^6 + 3844624*x^7 + 85700980*x^8 + 2011283640*x^9 + 49248127800*x^10 + 1250064156912*x^11 + 32736194249256*x^12 +... such that A(x - A(x)^2) = x + 3*A(x)^2. RELATED SERIES. Note that Series_Reversion(x - A(x)^2) = 3*x/4 + A(x)/4, which begins: Series_Reversion(x - A(x)^2) = x + x^2 + 10*x^3 + 141*x^4 + 2398*x^5 + 46002*x^6 + 961156*x^7 + 21425245*x^8 + 502820910*x^9 + 12312031950*x^10 + 312516039228*x^11 + 8184048562314*x^12 +... Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x, R(x) = x - 4*x^2 - 8*x^3 - 84*x^4 - 1112*x^5 - 17352*x^6 - 303824*x^7 - 5791060*x^8 - 117898648*x^9 - 2531645240*x^10 - 56835852080*x^11 - 1325547044072*x^12 +... then Series_Reversion(x + 3*A(x)^2) = x/4 + 3*R(x)/4.
m = 24; A[_] = 0;
Do[A[x_] = x + 4 A[3x/4 + A[x]/4]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
{a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - F^2) - 3*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
G.f.: A(x) = x + 4*x^2 + 56*x^3 + 1172*x^4 + 30248*x^5 + 892296*x^6 + 28951344*x^7 + 1010322900*x^8 + 37384819496*x^9 + 1452697058744*x^10 + 58872642043856*x^11 + 2475764515398568*x^12 +... such that A(x - 3*A(x)^2) = x + A(x)^2. RELATED SERIES. Note that Series_Reversion(x - 3*A(x)^2) = x/4 + 3*A(x)/4, which begins: Series_Reversion(x - 3*A(x)^2) = x + 3*x^2 + 42*x^3 + 879*x^4 + 22686*x^5 + 669222*x^6 + 21713508*x^7 + 757742175*x^8 + 28038614622*x^9 + 1089522794058*x^10 + 44154481532892*x^11 + 1856823386548926*x^12 +... Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x, R(x) = x - 4*x^2 - 24*x^3 - 372*x^4 - 7944*x^5 - 204168*x^6 - 5942256*x^7 - 189500916*x^8 - 6490281480*x^9 - 235609789368*x^10 - 8983294304784*x^11 - 357373688297448*x^12 +... then Series_Reversion(x + A(x)^2) = 3*x/4 + R(x)/4.
m = 22; A[_] = 0;
Do[A[x_] = x + 4A[x/4 + 3A[x]/4]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)
{a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x - 3*F^2) - F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
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