A368627
Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + x*(A(x)^3 - A(-x)^3)/2.
Original entry on oeis.org
1, 1, 3, 7, 40, 103, 723, 1941, 15060, 41382, 340657, 950061, 8132676, 22916139, 201684153, 572618987, 5145063940, 14692661910, 134152006842, 384852888898, 3559210821120, 10248531332559, 95777105998365, 276630878235275, 2607824127882204, 7551545042631558, 71714198513326425
Offset: 0
G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 40*x^4 + 103*x^5 + 723*x^6 + 1941*x^7 + 15060*x^8 + 41382*x^9 + 340657*x^10 + 950061*x^11 + 8132676*x^12 + ...
where A(x) is formed from the even bisection of A(x)^2 and the odd bisection of A(x)^3, as can be seen from the expansions
A(x)^2 = 1 + 2*x + 7*x^2 + 20*x^3 + 103*x^4 + 328*x^5 + 1941*x^6 + 6506*x^7 + 41382*x^8 + 142892*x^9 + 950061*x^10 + ...
A(x)^3 = 1 + 3*x + 12*x^2 + 40*x^3 + 198*x^4 + 723*x^5 + 3927*x^6 + 15060*x^7 + 86190*x^8 + 340657*x^9 + 2016195*x^10 + ...
so that the bisections of the above series are related by
(A(x) - A(-x))/2 = x*(A(x)^2 + A(-x)^2)/2, and
(A(x) + A(-x))/2 = 1 + x*(A(x)^3 - A(-x)^3)/2.
SPECIFIC VALUES.
The g.f. A(x) converges at the radius of convergence r, given by
A(-r) = 1 at r = (A(r) - 1)/(1 + A(r)^2) = 0.1795090246029167685576...
where A(r) = (1 + (28 + sqrt(783))^(1/3) + (28 - sqrt(783))^(1/3))/3 = 1.6956207695598620574163671... solves A(r)^3 - A(r)^2 = 2.
Other values are as follows.
A(t) = 3/2 at t = 0.1762576405478293392948378476047094214871919048852854...
with A(-t) = 0.9457634131178785046715685513829104426794138117773372...
A(1/6) = 1.39045291214794641706750008755820521981873579773148377...
A(-1/6) = 0.92547553450368274047514062093278734252641968691372863...
A(1/7) = 1.26282273990610251025800463852287012565418776197621997...
A(-1/7) = 0.91531855101291210815598364280272856428949318592006407...
A(1/8) = 1.20403758075277993770588254622742634950821058062345547...
A(-1/8) = 0.91758011120888933832570407861048171782335413914549218...
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{a(n) = my(A=1+x,B); for(i=1,n, A=truncate(A)+x*O(x^i); B=subst(A,x,-x);
A = 1 + x*(A^2 + B^2)/2 + x*(A^3 - B^3)/2 ; ); polcoeff(A,n)}
for(n=0, 30, print1(a(n), ", "))
A368629
Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 + A(-x)^2)/2 + x*(A(x)^4 - A(-x)^4)/2.
Original entry on oeis.org
1, 1, 4, 9, 88, 210, 2644, 6493, 91992, 229646, 3484008, 8789562, 139443168, 354379540, 5801987316, 14824740645, 248459660984, 637465292438, 10878564788984, 28001827694446, 484778825103504, 1251132971284668, 21915195896364296, 56682787977509650, 1002570518541796720
Offset: 0
G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 88*x^4 + 210*x^5 + 2644*x^6 + 6493*x^7 + 91992*x^8 + 229646*x^9 + 3484008*x^10 + 8789562*x^11 + 139443168*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 9*x^2 + 26*x^3 + 210*x^4 + 668*x^5 + 6493*x^6 + 21538*x^7 + 229646*x^8 + 779772*x^9 + 8789562*x^10 + ...
A(x)^4 = 1 + 4*x + 22*x^2 + 88*x^3 + 605*x^4 + 2644*x^5 + 20114*x^6 + 91992*x^7 + 741154*x^8 + 3484008*x^9 + 29125100*x^10 + ...
The odd bisection of A(x) may be formed from the even bisection of A(x)^2:
(A(x) - A(-x))/2 = x + 9*x^3 + 210*x^5 + 6493*x^7 + 229646*x^9 + ...
(A(x)^2 + A(-x)^2)/2 = 1 + 9*x^2 + 210*x^4 + 6493*x^6 + 229646*x^8 + ...
The even bisection of A(x) may be formed from the odd bisection of A(x)^4:
(A(x) + A(-x))/2 = 1 + 4*x^2 + 88*x^4 + 2644*x^6 + 91992*x^8 + 3484008*x^10 + ...
(A(x)^4 - A(-x)^4)/2 = 4*x + 88*x^3 + 2644*x^5 + 91992*x^7 + 3484008*x^9 + ...
SPECIFIC VALUES.
A(-r) = 1 and A(r) = sqrt(2) at r = (sqrt(2) - 1)/3 = 0.138071....
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{a(n) = my(A=1+x,B); for(i=1,n, A=truncate(A)+x*O(x^i); B=subst(A, x, -x);
A = 1 + x*(A^2 + B^2)/2 + x*(A^4 - B^4)/2 ; ); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
A368593
Expansion of g.f. A(x) satisfying A(x) = 1 + x*(A(x)^2 - A(-x)^2)/2 + x*sqrt( (A(x)^4 + A(-x)^4)/2 ).
Original entry on oeis.org
1, 1, 2, 7, 18, 78, 220, 1043, 3090, 15402, 47044, 242126, 755076, 3973820, 12580344, 67303139, 215511330, 1167556434, 3772175860, 20640707866, 67167649868, 370510806212, 1212836703304, 6735128062542, 22156120392276, 123731147310820, 408741630687656, 2293595176625340
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 18*x^4 + 78*x^5 + 220*x^6 + 1043*x^7 + 3090*x^8 + 15402*x^9 + 47044*x^10 + 242126*x^11 + 755076*x^12 + ...
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 54*x^4 + 220*x^5 + 717*x^6 + 3090*x^7 + 10562*x^8 + 47044*x^9 + 165858*x^10 + 755076*x^11 + ...
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 205*x^4 + 836*x^5 + 3178*x^6 + 13192*x^7 + 51490*x^8 + 216808*x^9 + 862588*x^10 + ...
The even bisection of A(x) may be formed from the odd bisection of A(x)^2:
(A(x) + A(-x))/2 = 1 + 2*x^2 + 18*x^4 + 220*x^6 + 3090*x^8 + 47044*x^10 + ...
(A(x)^2 - A(-x)^2)/2 = 2*x + 18*x^3 + 220*x^5 + 3090*x^7 + 47044*x^9 + ...
The odd bisection of A(x) may be formed from the even bisection of A(x)^4:
(A(x) - A(-x))/2 = x + 7*x^3 + 78*x^5 + 1043*x^7 + 15402*x^9 + ...
(A(x)^4 + A(-x)^4)/2 = 1 + 14*x^2 + 205*x^4 + 3178*x^6 + 51490*x^8 + ...
sqrt( (A(x)^4 + A(-x)^4)/2 ) = 1 + 7*x^2 + 78*x^4 + 1043*x^6 + 15402*x^8 + ...
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{a(n) = my(A=[1]); for(i=1,n,
A = Vec(1 + x*(Ser(A)^2 - subst(Ser(A)^2,x,-x))/2 + x*sqrt( (Ser(A)^4 + subst(Ser(A)^4,x,-x))/2 ) +x*O(x^#A) ) );A[n+1]}
for(n=0,30,print1(a(n),", "))
Showing 1-3 of 3 results.
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