A271931
G.f. A(x) satisfies: A(x) = A( x^5 + 5*x*A(x)^5 )^(1/5), with A(0)=0, A'(0)=1.
Original entry on oeis.org
1, 1, 3, 11, 44, 185, 803, 3564, 16082, 73502, 339391, 1580318, 7410356, 34956846, 165756814, 789543189, 3775883483, 18122280953, 87257629998, 421366007784, 2040186607333, 9902368905093, 48170863713973, 234819266573684, 1146894750998644, 5611743950271715, 27504683191546135
Offset: 1
G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 44*x^5 + 185*x^6 + 803*x^7 + 3564*x^8 + 16082*x^9 + 73502*x^10 + 339391*x^11 + 1580318*x^12 + ...
where A(x)^5 = A( x^5 + 5*x*A(x)^5 ).
RELATED SERIES.
A(x)^5 = x^5 + 5*x^6 + 25*x^7 + 125*x^8 + 625*x^9 + 3126*x^10 + 15640*x^11 + 78275*x^12 + 391875*x^13 + 1962500*x^14 + 9831253*x^15 + 49265695*x^16 + ...
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then R(x) begins
R(x) = x - x^2 - x^3 - x^4 + 4*x^6 + 15*x^7 + 36*x^8 + 55*x^9 - 359*x^11 - 1520*x^12 - 4028*x^13 - 6667*x^14 + 49062*x^16 + 217645*x^17 + ...
The 4 quintisections of R(x) = Q1(x) + Q2(x) + Q3(x) + Q4(x) (with the fifth being zero) are as follows
Q1(x) = x + 4*x^6 - 359*x^11 + 49062*x^16 - 8013396*x^21 + 1442958557*x^26 - 276352605126*x^31 + 55224710824185*x^36 - 11384289478228711*x^41 + ...
Q2(x) = -x^2 + 15*x^7 - 1520*x^12 + 217645*x^17 - 36405005*x^22 + 6650838668*x^27 - 1286179025729*x^32 + 258819346825534*x^37 + ...
Q3(x) = -x^3 + 36*x^8 - 4028*x^13 + 600254*x^18 - 102567034*x^23 + 18988120493*x^28 - 3705388523045*x^33 + 750546817970646*x^38 + ...
Q4(x) = -x^4 + 55*x^9 - 6667*x^14 + 1028514*x^19 - 179152944*x^24 + 33573744984*x^29 - 6607215559460*x^34 + 1346634048063165*x^39 + ...
where Q1*Q4 = -Q2*Q3 where
Q2*Q3 = x^5 - 51*x^10 + 6088*x^15 - 933039*x^20 + 161933629*x^25 - 30277104991*x^30 + 5949003081867*x^35 - 1211076410858363*x^40 + ...
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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^5 + 5*X*A^5)^(1/5) ); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))
A271932
G.f. A(x) satisfies: A(x) = A( x^7 + 7*x*A(x)^7 )^(1/7), with A(0)=0, A'(0)=1.
Original entry on oeis.org
1, 1, 4, 20, 110, 638, 3828, 23515, 146970, 930820, 5957325, 38452405, 249944939, 1634287025, 10739831400, 70884562683, 469622328252, 3121694320866, 20811920304961, 139115729296575, 932107335003790, 6258662787526655, 42105353650697301, 283765005631661148, 1915495724241980280, 12949332513585521217, 87661142189041380207, 594176943178375193748, 4032121696383579351905, 27392082325012470506385, 186276500908841717917320
Offset: 1
G.f.: A(x) = x + x^2 + 4*x^3 + 20*x^4 + 110*x^5 + 638*x^6 + 3828*x^7 + 23515*x^8 + 146970*x^9 + 930820*x^10 + 5957325*x^11 + 38452405*x^12 +...
where A(x)^7 = A( x^7 + 7*x*A(x)^7 ).
RELATED SERIES.
A(x)^7 = x^7 + 7*x^8 + 49*x^9 + 343*x^10 + 2401*x^11 + 16807*x^12 + 117649*x^13 + 823544*x^14 + 5764822*x^15 + 40353901*x^16 + 282478679*x^17 + 1977362758*x^18 + 13841640148*x^19 + 96892304579*x^20 + 678252720401*x^21 +...
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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^7 + 7*X*A^7)^(1/7) ); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))
A356780
Coefficients in the odd function A(x) such that: A(x) = A( x^2 + 2*x^2*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.
Original entry on oeis.org
1, 1, 2, 6, 21, 78, 303, 1223, 5085, 21623, 93585, 410894, 1825682, 8193544, 37087449, 169114547, 776110247, 3581944258, 16614576945, 77410877233, 362126147797, 1700179143293, 8008689767674, 37838553977426, 179268540549758, 851478474635404, 4053760582437106
Offset: 1
G.f. A(x) = x + x^3 + 2*x^5 + 6*x^7 + 21*x^9 + 78*x^11 + 303*x^13 + 1223*x^15 + 5085*x^17 + 21623*x^19 + 93585*x^21 + ...
where A(x)^2 = A( x^2 + 2*x^2*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^4 + 5*x^6 + 16*x^8 + 58*x^10 + 222*x^12 + 882*x^14 + 3616*x^16 + 15205*x^18 + 65220*x^20 + ...
x^2 + 2*x^2*A(x)^2 = x^2 + 2*x^4 + 4*x^6 + 10*x^8 + 32*x^10 + 116*x^12 + 444*x^14 + 1764*x^16 + 7232*x^18 + 30410*x^20 + ...
Let G(x) = Series_Reversion( A(x) ) then
G(x) = x - x^3 + x^5 - 2*x^7 + 4*x^9 - 7*x^11 + 12*x^13 - 23*x^15 + 45*x^17 - 84*x^19 + 157*x^21 - 302*x^23 + 584*x^25 - 1121*x^27 + ...
where G(x)^2 = G(x^2)/(1 + 2*x^2) and G(A(x)) = x.
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{a(n) = my(A=x+x^3, X=x+x*O(x^(2*n))); for(i=1, 2*n, A = subst(A, x, x^2 + 2*X^2*A^2)^(1/2) ); polcoeff(A, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))
Showing 1-3 of 3 results.
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