A264414
G.f. A(x) satisfies: A(x)^2 = A(x^2) + 20*x.
Original entry on oeis.org
1, 10, -45, 450, -5535, 75600, -1106100, 16953750, -268652880, 4365638550, -72354858300, 1218356280000, -20784495119850, 358457180010750, -6239532583193625, 109476057598087500, -1934128026918961515, 34378012275668994150, -614328464414815220025, 11030366153872043358750, -198899407327466712808800, 3600377821710426377668500
Offset: 0
G.f.: A(x) = 1 + 10*x - 45*x^2 + 450*x^3 - 5535*x^4 + 75600*x^5 - 1106100*x^6 +...
where
A(x)^2 = 1 + 20*x + 10*x^2 - 45*x^4 + 450*x^6 - 5535*x^8 + 75600*x^10 - 1106100*x^12 +...
so that A(x)^2 = A(x^2) + 20*x.
Let G(x) = Series_Reversion( x / (A(x^2) + 4*x) ), then
G(x) = x + 4*x^2 + 28*x^3 + 208*x^4 + 1702*x^5 + 14584*x^6 + 129808*x^7 + 1187008*x^8 + 11089153*x^9 + 105370660*x^10 +...+ A264226(n)*x^n +...
such that G(x)^2 = G( x^2/(1-8*x) ).
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{a(n) = my(A=1); for(i=1,n, A = sqrt( subst(A,x,x^2) + 20*x +x*O(x^n))); polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))
A271930
G.f. A(x) satisfies: A(x) = A( x^2 + 6*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.
Original entry on oeis.org
1, 3, 15, 90, 597, 4221, 31185, 237897, 1859568, 14816637, 119892942, 982565883, 8138777166, 68028775587, 573078135996, 4860507197700, 41470162208814, 355695498901179, 3065210379987489, 26525947283576640, 230425563258798840, 2008561878414115803, 17563090615911038115, 154014411705019299450, 1354142406561753259035, 11934928413519024726252, 105426063390991627937457, 933206579920813459523994, 8276480132736299734057275, 73535083052134446419214960
Offset: 1
G..f.: A(x) = x + 3*x^2 + 15*x^3 + 90*x^4 + 597*x^5 + 4221*x^6 + 31185*x^7 + 237897*x^8 + 1859568*x^9 + 14816637*x^10 + 119892942*x^11 + 982565883*x^12 +...
where A(x)^2 = A( x^2 + 6*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 39*x^4 + 270*x^5 + 1959*x^6 + 14724*x^7 + 113706*x^8 + 896994*x^9 + 7198257*x^10 + 58580766*x^11 + 482345937*x^12 + 4011023556*x^13 + 33637887441*x^14 +...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 3*x^2 + 3*x^3 - 3*x^5 + 9*x^7 - 33*x^9 + 126*x^11 - 513*x^13 + 2214*x^15 - 9876*x^17 + 45045*x^19 - 209493*x^21 +...+ A264412(n)*x^(2*n+1) +...
such that B(x) = x*G(x^2) - 3*x^2 where G(x)^2 = G(x^2) + 6*x, and G(x) is the g.f. of A264412.
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{a(n) = my(A=x,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 6*X*A^2)^(1/2) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
A271935
G.f. A(x) satisfies: A(x) = A( x^2 + 8*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.
Original entry on oeis.org
1, 4, 26, 200, 1691, 15204, 142710, 1382568, 13721765, 138802136, 1425785270, 14832383488, 155947271878, 1654494195340, 17690004381000, 190426309700616, 2062071992480208, 22447191471665160, 245501068961175090, 2696300196714320520, 29725402250477117175, 328835072363241763920
Offset: 1
G..f.: A(x) = x + 4*x^2 + 26*x^3 + 200*x^4 + 1691*x^5 + 15204*x^6 + 142710*x^7 + 1382568*x^8 + 13721765*x^9 + 138802136*x^10 + 1425785270*x^11 + ...
where A(x)^2 = A( x^2 + 8*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 8*x^3 + 68*x^4 + 608*x^5 + 5658*x^6 + 54336*x^7 + 534984*x^8 + 5373824*x^9 + 54866075*x^10 + 567775856*x^11 + 5942353444*x^12 + ...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 4*x^2 + 6*x^3 - 15*x^5 + 90*x^7 - 660*x^9 + 5310*x^11 - 45765*x^13 + 413640*x^15 - 3864345*x^17 + 37014120*x^19 + ... + A264413(n)*x^(2*n+1) + ...
such that B(x) = x*G(x^2) - 4*x^2 where G(x)^2 = G(x^2) + 12*x, and G(x) is the g.f. of A264413.
From _Paul D. Hanna_, May 20 2024: (Start)
The series (A(x)/x)^(1/4) seems to consist solely of integer coefficients
(A(x)/x)^(1/4) = 1 + x + 5*x^2 + 34*x^3 + 268*x^4 + 2305*x^5 + 20988*x^6 + 198891*x^7 + 1941111*x^8 + 19377707*x^9 + 196936775*x^10 + ...
and continues to be integral for at least the initial 400 coefficients. (End)
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{a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, x^2 + 8*X*A^2)^(1/2) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
Showing 1-3 of 3 results.
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