A227852
G.f. A(x) satisfies: A(x) = Series_Reversion( x - (A(x)^2 + A(-x)^2)/2 ).
Original entry on oeis.org
1, 1, 2, 10, 44, 294, 1728, 13389, 93130, 796620, 6235288, 57551130, 493813936, 4857378920, 44989814920, 468103507718, 4633862094852, 50749496457992, 533271010341720, 6126256486912776, 67990630238066888, 817168635245112432, 9541543704324657008, 119719059789052412360
Offset: 1
G.f.: A(x) = x + x^2 + 2*x^3 + 10*x^4 + 44*x^5 + 294*x^6 + 1728*x^7 +...
The series reversion of A(x), G(x) where A(G(x)) = x, begins:
G(x) = x - x^2 - 5*x^4 - 112*x^6 - 4320*x^8 - 227766*x^10 - 14942616*x^12 - 1162657840*x^14 +...+ (-1)^n * A263531(n)*x^(2*n) +...
and can be formed from a bisection of A(x)^2:
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 24*x^5 + 112*x^6 + 716*x^7 + 4320*x^8 + 32290*x^9 + 227766*x^10 + 1893488*x^11 + 14942616*x^12 + 134816212*x^13 + 1162657840*x^14 +...
The related g.f. of A263531, F(x) = -(A(I*x)^2 + A(-I*x)^2)/2, satisfies: F(x) = B(x)^2 - C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1:
F(x) = x^2 - 5*x^4 + 112*x^6 - 4320*x^8 + 227766*x^10 - 14942616*x^12 +...
-
{a(n)=local(A=x);for(i=1,n,A=serreverse(x-(A^2+subst(A^2,x,-x +x*O(x^n)))/2));polcoeff(A,n)}
for(n=1,25,print1(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+subst(A,x,-x)^2)^m/2^m/m!))+x*O(x^n)); polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))
A263530
G.f. A(x) satisfies: A(x) = B(x)^2 + C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*A(x)), where I^2 = -1.
Original entry on oeis.org
1, -3, 52, -1596, 68174, -3679964, 238949640, -18133397519, 1578639190316, -155623090726884, 17203681850199360, -2116171636238243028, 287762930191296817296, -43014624174283817327952, 7032470676704382424751408, -1251802142595596587066746328, 241602713767787669715442097616, -50368862903110844612768593045136, 11303387910446267256159298807620472
Offset: 1
G.f.: A(x) = x^2 - 3*x^4 + 52*x^6 - 1596*x^8 + 68174*x^10 - 3679964*x^12 + 238949640*x^14 - 18133397519*x^16 +...
such that A(x) = B(x)^2 + C(x)^2 and B(x) and C(x) are defined by
Series_Reversion(x - I*A(x)) = B(x) + I*C(x), where
B(x) = x - 2*x^3 + 32*x^5 - 944*x^7 + 39366*x^9 - 2090576*x^11 + 134136792*x^13 - 10085875720*x^15 + 871536657504*x^17 +...+ (-1)^(n-1)*A141202(2*n-1)*x^(2*n-1) +...
C(x) = x^2 - 8*x^4 + 178*x^6 - 6255*x^8 + 293652*x^10 - 17085798*x^12 + 1182991528*x^14 - 95087538324*x^16 +...+ (-1)^(n-1)*A141202(2*n)*x^(2*n) +...
and
B(x)^2 = x^2 - 4*x^4 + 68*x^6 - 2016*x^8 + 83532*x^10 - 4399032*x^12 + 280046448*x^14 - 20916418480*x^16 + 1797498262020*x^18 +...
C(x)^2 = x^4 - 16*x^6 + 420*x^8 - 15358*x^10 + 719068*x^12 - 41096808*x^14 + 2783020961*x^16 - 218859071704*x^18 +...
Further
G(x) = -I*B(I*x) - C(I*x) = x + x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 178*x^6 + 944*x^7 + 6255*x^8 + 39366*x^9 + 293652*x^10 +...+ A141202(n)*x^n +...
where G(x + G(x)*G(-x)) = x.
-
{a(n) = my(A=x^2, D); for(i=0,2*n, D=serreverse(x - I*A +O(x^(2*n+1))); A = real(D)^2 + imag(D)^2 ); polcoeff(A,2*n)}
for(n=1,20,print1(a(n),", "))
-
/* Differential Series */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A = x^2 +O(x^(2*n+2))); for(i=1, 2*n, D = x + sum(m=1, 2*n, I^m*Dx(m-1, A^m/m!) +O(x^(2*n+2))); A = real(D)^2 + imag(D)^2 ); polcoeff(A, 2*n)}
for(n=1, 20, print1(a(n), ", "))
A295767
G.f. A(x) satisfies: A(x + A(x)*A(-x)) = x - A(x)*A(-x).
Original entry on oeis.org
1, 2, 4, 18, 76, 500, 2888, 23018, 160556, 1449996, 11575640, 114832932, 1019757080, 10926139752, 106088136208, 1215141302498, 12753198909052, 155094128725196, 1745058840478104, 22420718376535948, 268759075046461512, 3634051693946151736, 46176378783947578800, 655022571579520952068, 8785797027703008422264, 130388708648538590304216, 1839515449214236524003120
Offset: 1
G.f.: A(x) = x + 2*x^2 + 4*x^3 + 18*x^4 + 76*x^5 + 500*x^6 + 2888*x^7 + 23018*x^8 + 160556*x^9 + 1449996*x^10 + 11575640*x^11 + 114832932*x^12 + 1019757080*x^13 + 10926139752*x^14 + 106088136208*x^15 + 1215141302498*x^16 +...
such that A(x + A(x)*A(-x)) = x - A(x)*A(-x).
RELATED SERIES.
-A(x)*A(-x) = x^2 + 4*x^4 + 96*x^6 + 4060*x^8 + 239920*x^10 + 17996072*x^12 + 1630314752*x^14 + 173202828908*x^16 + 21167253920784*x^18 + 2935439183937720*x^20 +...
(A(x) + x)/2 = x + x^2 + 2*x^3 + 9*x^4 + 38*x^5 + 250*x^6 + 1444*x^7 + 11509*x^8 + 80278*x^9 + 724998*x^10 + 5787820*x^11 + 57416466*x^12 +...
sqrt( -A(x)*A(-x) ) = x + 2*x^3 + 46*x^5 + 1938*x^7 + 115026*x^9 + 8678836*x^11 + 790630586*x^13 + 84398006438*x^15 + 10355026866054*x^17 + 1440596696075200*x^19 + 224937262867609220*x^21 +...
-
{a(n) = my(A=[1],G=x); for(i=1,n, A=concat(A,0); G = x*Ser(A); A[#A] = -Vec(subst(G,x, x + G*subst(G,x,-x)) + G*subst(G,x,-x))[#A]);A[n]}
for(n=1,30,print1(a(n),", "))
Showing 1-3 of 3 results.