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),", "))
A271957
G.f. A(x) satisfies: A(x) = A( x^2 + 10*x*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.
Original entry on oeis.org
1, 5, 40, 375, 3845, 41825, 474450, 5552250, 66548785, 812875800, 10082125950, 126637168125, 1607562407775, 20591392666250, 265810034489750, 3454516382881875, 45162288467005155, 593528625987396725, 7836767285955169200, 103908861022437312375, 1382961699685548183750, 18469547560714428659250, 247433242662040209056250, 3324296142183357299203125, 44779542961314348791789400, 604655933814703316140014375
Offset: 1
G..f.: A(x) = x + 5*x^2 + 40*x^3 + 375*x^4 + 3845*x^5 + 41825*x^6 + 474450*x^7 + 5552250*x^8 + 66548785*x^9 + 812875800*x^10 + 10082125950*x^11 + 126637168125*x^12 +...
where A(x)^2 = A( x^2 + 10*x*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 10*x^3 + 105*x^4 + 1150*x^5 + 13040*x^6 + 152100*x^7 + 1815375*x^8 + 22078750*x^9 + 272728845*x^10 + 3412891200*x^11 + 43178951325*x^12 +...
Let B(x) be the series reversion of the g.f. A(x), A(B(x)) = x, then:
B(x) = x - 5*x^2 + 10*x^3 - 45*x^5 + 450*x^7 - 5535*x^9 + 75600*x^11 - 1106100*x^13 + 16953750*x^15 +...+ A264414(n)*x^(2*n+1) +...
such that B(x) = x*G(x^2) - 5*x^2 where G(x)^2 = G(x^2) + 20*x, and G(x) is the g.f. of A264414.
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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 + 10*X*A^2)^(1/2) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
A377106
G.f. A(x) satisfies A(x)^3 = A( x^3 + 9*x*A(x)^3 ), with A(0)=0, A'(0)=1.
Original entry on oeis.org
1, 3, 18, 127, 966, 7686, 63068, 529503, 4526262, 39262658, 344789172, 3059733222, 27400769364, 247345475628, 2248572742200, 20570124766951, 189238723449318, 1749776993081730, 16253403563598516, 151604206816149210, 1419457992097097340, 13336331712054463644, 125697697304515725840
Offset: 1
G.f.: A(x) = x + 3*x^2 + 18*x^3 + 127*x^4 + 966*x^5 + 7686*x^6 + 63068*x^7 + 529503*x^8 + 4526262*x^9 + 39262658*x^10 + 344789172*x^11 + 3059733222*x^12 + ...
where A(x)^3 = A( x^3 + 9*x*A(x)^3 ).
RELATED SERIES.
A(x)^3 = x^3 + 9*x^4 + 81*x^5 + 732*x^6 + 6642*x^7 + 60507*x^8 + 553329*x^9 + 5079024*x^10 + 46788678*x^11 + 432520930*x^12 + ...
Series reversion of A(x) equals B(x) - 3*x^3/B(x) where
B(x) = x + 8*x^4 - 280*x^7 + 15328*x^10 - 1007576*x^13 + 73169608*x^16 - 5656895520*x^19 + 456585800584*x^22 - 38029012055320*x^25 + 3244225801946920*x^28 - 282033503420822552*x^31 + ...
so that A( B(x) - 3*x^3/B(x) ) = x.
SPECIFIC VALUES.
A(t) = 1/6 at t = 0.0913017665091460949496315519875858022728583060252844...
where 1/216 = A( t^3 + t/24 ).
A(t) = 1/9 at t = 0.0756231400530157002966336216229658355706050775929719...
where 1/729 = A( t^3 + t/81 ).
A(1/11) = 0.16461186433566159924255427988603576152486558542514...
A(1/12) = 0.13356888809515041673070959997705841146178687774042...
A(1/13) = 0.11450357672473104104332015691591377007745191359804...
A(1/15) = 0.09064971528132540512370615784788517775098854995359...
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{a(n) = my(A=x+3*x^2); for(m=1, n, A = truncate(A); A = subst(A, x, x^3 + 9*x*A^3 +x^4*O(x^m))^(1/3) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
Showing 1-4 of 4 results.
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