A171991
G.f. satisfies: A(x) = x + A(A(x))^2 + A(A(x))^4.
Original entry on oeis.org
1, 1, 4, 25, 190, 1644, 15636, 159977, 1736392, 19804646, 235758596, 2914539808, 37275052828, 491727695628, 6675266957376, 93073877729749, 1330851640325930, 19490214361291636, 292025342161401292, 4472427699080578554, 69958787931298226280, 1116911386287712897260
Offset: 1
G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1644*x^6 + 15636*x^7 +...
Related series begin:
A(A(x)) = x + 2*x^2 + 10*x^3 + 71*x^4 + 598*x^5 + 5634*x^6 + 57624*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 182*x^5 + 1580*x^6 + 15080*x^7 +...
A(A(x))^4 = x^4 + 8*x^5 + 64*x^6 + 556*x^7 + 5192*x^8 + 51536*x^9 +...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 58*x^5 + 446*x^6 + 3868*x^7 + 36705*x^8 +...
A(x)^4 = x^4 + 4*x^5 + 22*x^6 + 152*x^7 + 1205*x^8 + 10564*x^9 +...
where the series reversion of the g.f. A(x) begins:
x - A(x)^2 - A(x)^4 = x - x^2 - 2*x^3 - 10*x^4 - 62*x^5 - 468*x^6 - 4020*x^7 -...
-
terms = 23; A[] = 0; Do[A[x] = x + A[A[x]]^2 + A[A[x]]^4 + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, May 04 2025 *)
-
{a(n)=local(A=x+x^2,B=x); for(i=1, n, B=subst(A,x,A+x*O(x^n));A=x+B^2+B^4); polcoeff(A, n)}
-
{a(n)=local(A=x+x^2); for(i=1, n, A=serreverse(x-A^2-A^4+x*O(x^n))); polcoeff(A, n)}
for(n=1,30,print1(a(n),", "))
A211794
G.f. satisfies: A(x) = x + A(A(x))^2 + A(A(x))^3.
Original entry on oeis.org
1, 1, 5, 34, 284, 2698, 28116, 314558, 3726504, 46310523, 599691087, 8051875596, 111674902936, 1595269110991, 23416737953975, 352551682694096, 5435771974479840, 85722307390977058, 1381216396128765272, 22718307467700650259, 381156536404199989205
Offset: 1
G.f.: A(x) = x + x^2 + 5*x^3 + 34*x^4 + 284*x^5 + 2698*x^6 + 28116*x^7 + ...
Related series begin:
A(A(x)) = x + 2*x^2 + 12*x^3 + 94*x^4 + 872*x^5 + 9026*x^6 + 101194*x^7 + ...
A(A(x))^2 = x^2 + 4*x^3 + 28*x^4 + 236*x^5 + 2264*x^6 + 23796*x^7 + ...
A(A(x))^3 = x^3 + 6*x^4 + 48*x^5 + 434*x^6 + 4320*x^7 + 46302*x^8 + ...
A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 78*x^5 + 661*x^6 + 6304*x^7 + 65624*x^8 + ...
A(x)^3 = x^3 + 3*x^4 + 18*x^5 + 133*x^6 + 1146*x^7 + 10995*x^8 + ...
where the series reversion R(x) of the g.f. A(x) begins:
R(x) = x - A(x)^2 - A(x)^3 = x - x^2 - 3*x^3 - 14*x^4 - 96*x^5 - 794*x^6 - 7450*x^7 - 76619*x^8 - 846161*x^9 - 9901282*x^10 + ...
Also, the series reversion of A(A(x)) is given by
x - x^2 - x^3 - A(x)^2 - A(x)^3 = x - 2*x^2 - 4*x^3 - 14*x^4 - 96*x^5 - 794*x^6 - 7450*x^7 - 76619*x^8 - 846161*x^9 - 9901282*x^10 - ...
Further, the series reversion of A(A(A(x))) starts as
x - x^2 - x^3 - A(x)^2 - A(x)^3 - R(x)^2 - R(x)^3 = x - 3*x^2 - 3*x^3 - 6*x^4 - 68*x^5 - 614*x^6 - 5952*x^7 - 62456*x^8 - 699438*x^9 - 8270469*x^10 + ...
-
{a(n)=local(A=x+x^2, B=x); for(i=1, n, B=subst(A, x, A+x*O(x^n)); A=x+B^2+B^3); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
-
{a(n)=local(A=x+x^2); for(i=1, n, A=serreverse(x-A^2-A^3+x*O(x^n))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
A277311
G.f. satisfies: A(x - 5*A(x)^2) = x - 4*A(x)^2.
Original entry on oeis.org
1, 1, 12, 200, 4034, 92752, 2353272, 64579809, 1891598860, 58591554652, 1906271367296, 64816527248936, 2294331974613872, 84290267670946720, 3206227129084419920, 126022120854865417140, 5110001578581607976400, 213458728365617240931360, 9175021814527973211291880, 405366362599820848509766760, 18392202994173383123235536800, 856255190568423353781484124240
Offset: 1
G.f.: A(x) = x + x^2 + 12*x^3 + 200*x^4 + 4034*x^5 + 92752*x^6 + 2353272*x^7 + 64579809*x^8 + 1891598860*x^9 + 58591554652*x^10 +...
such that A(x - 5*A(x)^2) = x - 4*A(x)^2.
A(x)^2 = x^2 + 2*x^3 + 25*x^4 + 424*x^5 + 8612*x^6 + 198372*x^7 + 5028864*x^8 + 137705810*x^9 + 4022209822*x^10 + 124205854376*x^11 + 4028545272136*x^12 + 136566005356212*x^13 + 4820263259998720*x^14 + 176614868022441920*x^15 +...
A(x - 5*A(x)^2) = x - 4*x^2 - 8*x^3 - 100*x^4 - 1696*x^5 - 34448*x^6 - 793488*x^7 - 20115456*x^8 - 550823240*x^9 - 16088839288*x^10 +...
which equals x - 4*A(x)^2.
Series_Reversion(x - 5*A(x)^2) = x + 5*x^2 + 60*x^3 + 1000*x^4 + 20170*x^5 + 463760*x^6 + 11766360*x^7 + 322899045*x^8 + 9457994300*x^9 +...
which equals 5*A(x) - 4*x.
A( 5*A(x) - 4*x ) = x + 6*x^2 + 82*x^3 + 1525*x^4 + 33864*x^5 + 848402*x^6 + 23259832*x^7 + 685028874*x^8 + 21411099560*x^9 + 704295189492*x^10 +24234549363096*x^11 + 868423052983416*x^12 + 32296557071230392*x^13 + 1243216715481216720*x^14 + 49428242214109804120*x^15 +...
which equals sqrt( A(x) -x ).
Cf.
A277300,
A277301,
A277302,
A277303,
A277304,
A277305,
A277306,
A277307,
A277308,
A277309,
A277310.
-
{a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-5*F^2) + 4*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))
A373314
Expansion of g.f. A(x) satisfying A(A(x)) - 4*A(A(A(x)))^2 = x.
Original entry on oeis.org
1, 2, 20, 304, 5728, 123680, 2942016, 75356544, 2048446208, 58517294080, 1744472116224, 53991571224576, 1728111953805312, 57027260271980544, 1935586663121272832, 67440373642584637440, 2408328339125296824320, 88029604711420113190912, 3289877540493975587913728, 125591805077248068782129152
Offset: 1
G.f.: A(x) = x + 2*x^2 + 20*x^3 + 304*x^4 + 5728*x^5 + 123680*x^6 + 2942016*x^7 + 75356544*x^8 + 2048446208*x^9 + 58517294080*x^10 + ...
where A(A(x)) - 4*A(A(A(x)))^2 = x.
RELATED SERIES.
A(A(x)) = x + 4*x^2 + 48*x^3 + 816*x^4 + 16704*x^5 + 385600*x^6 + 9705728*x^7 + 261167104*x^8 + ...
A(A(A(x))) = x + 6*x^2 + 84*x^3 + 1584*x^4 + 35168*x^5 + 869152*x^6 + 23222336*x^7 + 659257728*x^8 + ...
A(A(A(x)))^2 = x^2 + 12*x^3 + 204*x^4 + 4176*x^5 + 96400*x^6 + 2426432*x^7 + 65291776*x^8 + ...
Let B(x) be the series reversion of A(A(x)), B( A(A(x)) ) = x, then
B(x) = x - 4*A(x)^2 = x - 4*x^2 - 16*x^3 - 176*x^4 - 2752*x^5 - 52288*x^6 - 1129728*x^7 - 26801152*x^8 - ...
-
/* Using x = A(A(x)) - 4*A(A(A(x)))^2 */
{a(n) = my(A = [0,1],A1,A2,A3); for(i=1,n, A = concat(A,0); A1 = Ser(A);
A2 = subst(A1,x,A1); A3 = subst(A1,x,A2);
A[#A] = (1/2)*polcoeff(x - A2 + 4*A3^2, #A-1)); A[n+1]}
for(n=1,30,print1(a(n),", "))
A380678
G.f. A(x) satisfies A( x - A(x)^2/(1 - A(x)^2) ) = x.
Original entry on oeis.org
1, 1, 4, 25, 190, 1645, 15652, 160186, 1739032, 19838179, 236192158, 2920269202, 37352521348, 492799406899, 6690428699026, 93293086422514, 1334088426585850, 19538994465481000, 292775222237716612, 4484180296611470218, 70146488080451823382, 1119964903188050808163, 18239593214541431577550
Offset: 1
G.f.: A(x) = x + x^2 + 4*x^3 + 25*x^4 + 190*x^5 + 1645*x^6 + 15652*x^7 + 160186*x^8 + 1739032*x^9 + 19838179*x^10 + ...
where A(x - A(x)^2/(1 - A(x)^2)) = x.
Let B(x) = x - Series_Reversion(A(x)), where
B(x) = x^2 + 2*x^3 + 10*x^4 + 62*x^5 + 469*x^6 + 4028*x^7 + 37984*x^8 + 385202*x^9 + 4144798*x^10 + ... + A380558(n)*x^n + ...
then A(x) = x + B(A(x)) and B(x - B(x)) = x^2/(1 - x^2).
A171992
G.f. satisfies: A(x) = x + A(A(x))^2 - A(A(x))^4.
Original entry on oeis.org
1, 1, 4, 23, 166, 1380, 12684, 125857, 1328084, 14754242, 171338020, 2069009164, 25877555908, 334197713580, 4445788022944, 60800921601639, 853479846713406, 12280659254071964, 180929894848439516, 2726751302240331150, 42001984460083899448, 660800371941797598828
Offset: 1
G.f.: A(x) = x + x^2 + 4*x^3 + 23*x^4 + 166*x^5 + 1380*x^6 + 12684*x^7 +...
Related series begin:
A(A(x)) = x + 2*x^2 + 10*x^3 + 67*x^4 + 538*x^5 + 4866*x^6 + 48000*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 24*x^4 + 174*x^5 + 1444*x^6 + 13224*x^7 +...
A(A(x))^4 = x^4 + 8*x^5 + 64*x^6 + 540*x^7 + 4856*x^8 + 46352*x^9 +...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 54*x^5 + 394*x^6 + 3276*x^7 + 29985*x^8 +...
A(x)^4 = x^4 + 4*x^5 + 22*x^6 + 144*x^7 + 1085*x^8 + 9100*x^9 +...
where the series reversion of the g.f. A(x) begins:
x - A(x)^2 + A(x)^4 = x - x^2 - 2*x^3 - 8*x^4 - 50*x^5 - 372*x^6 - 3132*x^7 -...
-
{a(n)=local(A=x+x^2,B=x); for(i=1, n, B=subst(A,x,A+x*O(x^n));A=x+B^2-B^4); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
-
{a(n)=local(A=x+x^2); for(i=1, n, A=serreverse(x-A^2+A^4+x*O(x^n))); polcoeff(A, n)}
for(n=1,30,print1(a(n),", "))
A220379
G.f. satisfies: A(x) = x + (1-x)*A(A(x))^2.
Original entry on oeis.org
1, 1, 3, 16, 108, 836, 7136, 65708, 643522, 6638880, 71649726, 804833052, 9373104396, 112833093984, 1400634016520, 17894022203641, 234907670711601, 3164596264280695, 43700481254733535, 617995260359761384, 8942391804824517624, 132304132524112742604
Offset: 1
G.f.: A(x) = x + x^2 + 3*x^3 + 16*x^4 + 108*x^5 + 836*x^6 + 7136*x^7 +...
The g.f. satisfies the series:
A(x) = x + (1-x)*A(x)^2 + (1-x)^2*d/dx A(x)^4/2! + (1-x)^3*d^2/dx^2 A(x)^6/3! + (1-x)^4*d^3/dx^3 A(x)^8/4! +...
as well as the logarithmic series:
log(A(x)/x) = (1-x)*A(x)^2/x + (1-x)^2*[d/dx A(x)^4/x]/2! + (1-x)^3*[d^2/dx^2 A(x)^6/x]/3! + (1-x)^4*[d^3/dx^3 A(x)^8/x]/4! +...
Related expansions:
A(A(x)) = x + 2*x^2 + 8*x^3 + 48*x^4 + 354*x^5 + 2958*x^6 + 27004*x^7 +...
A(A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 128*x^5 + 964*x^6 + 8100*x^7 +...
(A(x)-x)/(1-x) = x^2 + 4*x^3 + 20*x^4 + 128*x^5 + 964*x^6 + 8100*x^7 +...
The series reversion of the g.f. A(x) equals:
(x-A(x)^2)/(1-A(x)^2) = x - x^2 - x^3 - 6*x^4 - 34*x^5 - 234*x^6 - 1818*x^7 -...
The series reversion of A(A(x)) equals:
1 - 1/((1+x)*(1-A(x)^2)) = x - 2*x^2 - 8*x^4 - 34*x^5 - 242*x^6 - 1852*x^7 -...
-
{a(n)=local(A=x);for(i=1,n,A=x+(1-x)*subst(A,x,A+x*O(x^n))^2);polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))
-
{a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse((x - A^2)/(1-A^2+x*O(x^n)))); 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, (1-x)^m*Dx(m-1, A^(2*m))/m!)+x*O(x^n)); 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*exp(sum(m=1, n, (1-x)^m*Dx(m-1, A^(2*m)/x)/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))
A291421
G.f. A(x) satisfies: A( x - A(x)^2*(1 + A(x))^2 ) = x.
Original entry on oeis.org
1, 1, 6, 45, 414, 4310, 49068, 598253, 7707738, 103981222, 1459259444, 21201220726, 317718863636, 4897066444332, 77455837982360, 1254882911977597, 20793816009974054, 351973815700006842, 6079707258590589100, 107070921557974264470, 1921112466081500096044, 35095122874748021511252, 652393778217784214993656, 12334667847853804120010726
Offset: 1
G.f.: A(x) = x + x^2 + 6*x^3 + 45*x^4 + 414*x^5 + 4310*x^6 + 49068*x^7 + 598253*x^8 + 7707738*x^9 + 103981222*x^10 + 1459259444*x^11 + 21201220726*x^12 + 317718863636*x^13 + 4897066444332*x^14 + 77455837982360*x^15 + 1254882911977597*x^16 +...
such that A( x - A(x)^2 - 2*A(x)^4 - A(x)^6 ) = x.
RELATED SERIES.
The g.f. of A185898 equals G(x) = A(x) + A(x)^2, which begins:
A(x) + A(x)^2 = x + 2*x^2 + 8*x^3 + 58*x^4 + 516*x^5 + 5264*x^6 + 59056*x^7 + 712002*x^8 + 9091360*x^9 + 121741316*x^10 +...+ A185898(n)*x^n +...
and satisfies G(x - G(x)^2) = x + x^2.
Also, we have the series:
A(x)^2*(1 + A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 148*x^5 + 1328*x^6 + 13520*x^7 + 150788*x^8 + 1804308*x^9 + 22852504*x^10 + 303523048*x^11 + 4199277144*x^12 +...
where A( x - A(x)^2*(1 + A(x))^2 ) = x.
Define the series reversion Ai(x) by Ai(A(x)) = x, then Ai(x) begins:
Ai(x) = x - x^2 - 4*x^3 - 20*x^4 - 148*x^5 - 1328*x^6 - 13520*x^7 - 150788*x^8 - 1804308*x^9 - 22852504*x^10 +...
so that Ai(x) = x - A(x)^2*(1 + A(x))^2.
Finally, another series of interest is
sqrt(A(x) - x) = x + 3*x^2 + 18*x^3 + 153*x^4 + 1534*x^5 + 17178*x^6 + 208276*x^7 + 2685135*x^8 + 36381426*x^9 + 513935734*x^10 + 7526074612*x^11 + 113767244374*x^12 + 1769506176124*x^13 + 28247513919396*x^14 + 461885675312008*x^15 + 7723529901763157*x^16 +...
-
{a(n) = my(A=x,V=[1]); for(i=1,n, V=concat(V,0); A=x*Ser(V); V[#V]=-polcoeff(subst(A,x, x - A^2*(1+A)^2),#V));V[n]}
for(n=1,30,print1(a(n),", "))
A376233
G.f. A(x) satisfies: A( x - x^2 - A(x)^3 ) = x.
Original entry on oeis.org
1, 1, 3, 13, 68, 401, 2576, 17670, 127786, 965890, 7583944, 61576198, 515209694, 4430434354, 39073275005, 352812956426, 3257141096415, 30708375193969, 295384611397245, 2896520002777988, 28934822132722103, 294279196616806638, 3045540859482010041, 32057787741226132890
Offset: 1
G.f.: A(x) = x + x^2 + 3*x^3 + 13*x^4 + 68*x^5 + 401*x^6 + 2576*x^7 + 17670*x^8 + 127786*x^9 + 965890*x^10 + 7583944*x^11 + 61576198*x^12 + ...
where A(x - x^2 - A(x)^3) = x.
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 58*x^6 + 318*x^7 + 1911*x^8 + 12330*x^9 + 84273*x^10 + 604503*x^11 + ...
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 171*x^6 + 1016*x^7 + 6531*x^8 + 44666*x^9 + 321418*x^10 + ...
A(A(x))^3 = x^3 + 6*x^4 + 36*x^5 + 230*x^6 + 1560*x^7 + 11139*x^8 + 83120*x^9 + 644472*x^10 + ...
where A(A(x))^3 = A(x) - x - A(x)^2.
A(A(x)) = x + 2*x^2 + 8*x^3 + 42*x^4 + 256*x^5 + 1721*x^6 + 12424*x^7 + 94796*x^8 + 756680*x^9 + ...
where A(A(y)) = x at y = x - 2*x^2 + x^3 - x^4 - (1 - 2*x + 2*x^2)*A(x)^3 - A(x)^6.
-
{a(n) = my(A=x); if(n<1, 0, for(i=1, n, A = serreverse(x - x^2 - A^3 + x*O(x^n))); polcoeff(A, n))}
for(n=1, 25, print1(a(n), ", "))
-
{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, (x^2 + A^3)^m)/m! +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
-
{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1, n, A = x*exp(sum(m=1, n, Dx(m-1, (x^2 + A^3)^m/x)/m!) +x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
A384622
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x) * A(x*A(x))^5 ).
Original entry on oeis.org
1, 1, 7, 75, 989, 14822, 242833, 4253818, 78573475, 1516124048, 30358711661, 627789264431, 13357722853019, 291611321803145, 6517101781199460, 148833150175812360, 3468184751644757228, 82363850033966966043, 1991430772785525516280, 48980124394583747435367
Offset: 0
-
a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*a(n-j, 5*j)));
Comments