A213591 -id:A213591 - cp's OEIS frontend

A212411 G.f. satisfies: A(x) = 1 + x*A(1 - 1/A(x))^2.

1, 1, 2, 7, 36, 235, 1792, 15261, 141382, 1401334, 14694166, 161714217, 1857003186, 22152227989, 273573165626, 3488210643709, 45820081884234, 618950367384072, 8585324020132250, 122127635117014779, 1779763238159032068, 26545963246376545934

Offset: 0

Paul D. Hanna, May 15 2012

nonn

Compare g.f. to the identity: G(x) = 1 + x*G(1-1/G(x)) when G(x) = 1/(1-x).

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 235*x^5 + 1792*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 90*x^4 + 570*x^5 + 4247*x^6 +...
1 - 1/A(x) = x + x^2 + 4*x^3 + 23*x^4 + 161*x^5 + 1286*x^6 + 11321*x^7 +...
A(1-1/A(x)) = 1 + x + 3*x^2 + 15*x^3 + 98*x^4 + 753*x^5 + 6471*x^6 +...
Let F(x) = A(1-1/A(x)), then F(1-1/A(x)) = A(1-1/F(x)):
F(1-1/A(x)) = 1 + x + 4*x^2 + 25*x^3 + 193*x^4 + 1693*x^5 + 16240*x^6 +...
...
Let G(x) be the g.f. of A213591, then
G(x) satisfies: x = G(x - G(x)^2) and G(x) = A(G(x)^2/x), where:
G(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
G(x)^2/x = x + 2*x^2 + 9*x^3 + 56*x^4 + 420*x^5 + 3572*x^6 +...
1/(1-G(x)^2/x) = 1 + x + 3*x^2 + 14*x^3 + 85*x^4 + 616*x^5 + 5072*x^6 +...
such that A(x/(1 - G(x)^2/x)) = 1/(1 - G(x)^2/x).

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A213591, A213628.

{a(n)=local(A=1+x);for(i=1,n,A=1+x*subst(A^2,x,1-1/(A+x*O(x^n))));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

Given g.f. A(x), let G(x) be the g.f. of A213591, then:
(1) G(x) = x*A(G(x)^2/x),
(2) A( x/(1 - G(x)^2/x) ) = 1/(1 - G(x)^2/x),
(3) x = G(x - G(x)^2).
G.f.: A(x) = x/Series_Reversion(F(x)) where F(x) = 1 - (x^2/F(x))/F(x^2/F(x)) is the g.f. of A213628.
Given g.f. A(x), let F(x) = A(1-1/A(x)), then F(1-1/A(x)) = A(1-1/F(x)) and A(x) = 1 + x*F(x)^2.

A276370 G.f. A(x) satisfies: A( x - A(x) ) = x^2.

Original entry on oeis.org

1, 2, 9, 56, 420, 3572, 33328, 334354, 3559310, 39838760, 465743720, 5658983108, 71191948512, 924554859776, 12365546196641, 169995491295312, 2398380272232272, 34680290150700800, 513390937937217088, 7773229533145403728, 120277760289804227632, 1900583166564027019136, 30649888151334972466392, 504153517331248726221392, 8454018409655883681321232, 144451967918022160558965408

Offset: 2

Paul D. Hanna, Sep 09 2016

nonn

An unsigned version of A138740 (apart from initial term).

Self-convolution of A213591.

			G.f.: A(x) = x^2 + 2*x^3 + 9*x^4 + 56*x^5 + 420*x^6 + 3572*x^7 + 33328*x^8 + 334354*x^9 + 3559310*x^10 + 39838760*x^11 + 465743720*x^12 +...
such that A( x - A(x) ) = x^2.

Paul D. Hanna, Table of n, a(n) for n = 2..302

Cf. A138740, A213591, A212411, A213628.

{a(n) = local(A=x^2); for(i=1, n, A = serreverse(x - A +x*O(x^n))^2); polcoeff(A, n)}
for(n=2,30,print1(a(n),", "))

{Dx(n, F) = local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = local(A=x^2 +x*O(x^n)); for(i=1, n, A = (x + sum(m=1, n, Dx(m-1, A^m)/m!) +x*O(x^n))^2); polcoeff(A, n)}
for(n=2,30,print1(a(n),", "))

G.f. satisfies:
(1) A(x) = ( x + A( sqrt(A(x)) ) )^2.
(2) A(x) = ( x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n / n! )^2.
(3) A(x) = x^2 * exp( 2*Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^n/x / n! ).
(4) A(x) = x^2 * G( A(x)/x )^2 where G(x) = 1 + x*G( 1 - 1/G(x) )^2 is the g.f. of A212411.
(5) A(x) = x^2 * F( sqrt(A(x)) )^2 where F(x) = 1 - (x^2/F(x)) / F( x^2/F(x) ) is the g.f. of A213628.

A213639 G.f. A(x) satisfies x = A( x - A(x)^3/x ).

Original entry on oeis.org

1, 1, 5, 38, 357, 3832, 45189, 572378, 7676653, 107971691, 1581714400, 24012849880, 376361077578, 6071985730614, 100602798234000, 1708558136679750, 29698002444820760, 527661478169200755, 9573199146196780335, 177192815265794698364, 3343432166097650920872

Offset: 1

Paul D. Hanna, Jun 16 2012

nonn

			G.f.: A(x) = x + x^2 + 5*x^3 + 38*x^4 + 357*x^5 + 3832*x^6 + 45189*x^7 +...
Related series:
A(x)^3/x = x^2 + 3*x^3 + 18*x^4 + 145*x^5 + 1389*x^6 + 14967*x^7 +...
A(A(x)) = x + 2*x^2 + 12*x^3 + 102*x^4 + 1042*x^5 + 11977*x^6 + 149776*x^7 +...
A(A(x))^3/A(x) = x^2 + 5*x^3 + 38*x^4 + 357*x^5 + 3832*x^6 + 45189*x^7 + ...
The g.f. satisfies:
A(x) = x + A(x)^3/x + [d/dx A(x)^6/x^2]/2! + [d^2/dx^2 A(x)^9/x^3]/3! + [d^3/dx^3 A(x)^12/x^4]/4! +...
Logarithmic series:
log(A(x)/x) = A(x)^3/x^2 + [d/dx A(x)^6/x^3]/2! + [d^2/dx^2 A(x)^9/x^4]/3! + [d^3/dx^3 A(x)^12/x^5]/4! +...
From _Seiichi Manyama_, Jun 05 2025: (Start)
Let b(n,k) = [x^n] (A(x)/x)^k.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,3*j).
a(n) = b(n-1,1). (End)

Cf. A213591.

{a(n)=local(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^3/x+x*O(x^n))); polcoeff(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, Dx(m-1, A^(3*m)/x^m)/m!)+x*O(x^n)); polcoeff(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, Dx(m-1, A^(3*m)/x^(m+1))/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 21, print1(a(n), ", "))

b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 3*j)));
a(n) = b(n-1, 1); \\ Seiichi Manyama, Jun 05 2025

G.f. satisfies:
(1) A(x) = x + A(A(x))^3 / A(x).
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x^n / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(3*n)/x^(n+1) / n! ).

A277310 G.f. satisfies: A(x - 4A(x)^2) = x - 3A(x)^2.

Original entry on oeis.org

1, 1, 10, 141, 2422, 47562, 1031764, 24214405, 606444990, 16055089470, 446238074892, 12955112773554, 391332183548956, 12261884937532340, 397576302315045800, 13313017677172350965, 459635990935574444942, 16339309997761322057206, 597340515437542895494748, 22435278085988347895795526, 864900964565994975048855444, 34195693888939483596581262668, 1385553440866978431053220575128

Offset: 1

Paul D. Hanna, Oct 12 2016

nonn

			G.f.: A(x) = x + x^2 + 10*x^3 + 141*x^4 + 2422*x^5 + 47562*x^6 + 1031764*x^7 + 24214405*x^8 + 606444990*x^9 + 16055089470*x^10 +...
such that A(x - 4*A(x)^2) = x - 3*A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 21*x^4 + 302*x^5 + 5226*x^6 + 102788*x^7 + 2226973*x^8 + 52126582*x^9 + 1301232638*x^10 + 34328704796*x^11 + 950803699394*x^12 + 27510261070028*x^13 + 828332416917876*x^14 + 25876801064095496*x^15 + 836682915170627501*x^16 +...
A(x - 4*A(x)^2) = x - 3*x^2 - 6*x^3 - 63*x^4 - 906*x^5 - 15678*x^6 - 308364*x^7 - 6680919*x^8 - 156379746*x^9 - 3903697914*x^10 +...
which equals x - 3*A(x)^2.
Series_Reversion(x - 4*A(x)^2) = x + 4*x^2 + 40*x^3 + 564*x^4 + 9688*x^5 + 190248*x^6 + 4127056*x^7 + 96857620*x^8 + 2425779960*x^9 + 64220357880*x^10 +...
which equals -3*x + 4*A(x).
A( 4*A(x) - 3*x ) = x + 5*x^2 + 58*x^3 + 921*x^4 + 17494*x^5 + 374994*x^6 + 8793460*x^7 + 221393569*x^8 + 5912166718*x^9 + 166058455158*x^10 + 4876311925036*x^11 + 149037482367530*x^12 + 4724877954111836*x^13 + 154959634972646340*x^14 + 5246331138228520168*x^15 +...
which equals  sqrt( A(x) - x ).

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A277295, A213591, A275765, A276360, A276361, A276362, A276363.

Cf. A277300, A277301, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309.

{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-4*F^2) + 3*F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) also satisfies:
(1) A(x) = x + A( 4*A(x) - 3*x )^2.
(2) A(x) = 3*x/4 + 1/4 * Series_Reversion(x - 4*A(x)^2).
(3) R(x) = 4*x/3 - 1/3 * Series_Reversion(x - 3*A(x)^2), where R(A(x)) = x.
(4) R( sqrt( x - R(x) ) ) = 4*x - 3*R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 4^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
a(n) = Sum_{k=0..n-1} A277295(n,k) * 4^k.

A360578 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion( x - xA'(x)A(x) ).

Original entry on oeis.org

1, 1, 5, 42, 471, 6422, 101439, 1803949, 35459549, 760744865, 17651187689, 439893743313, 11711735210140, 331666197753372, 9954249177284539, 315638779480717743, 10545365878475964736, 370309787453143694246, 13637805276205022293179, 525684316153586923528166

Offset: 1

Paul D. Hanna, Feb 21 2023

nonn

			G.f.: A(x) = x + x^2 + 5*x^3 + 42*x^4 + 471*x^5 + 6422*x^6 + 101439*x^7 + 1803949*x^8 + 35459549*x^9 + 760744865*x^10 + ...
such that A( x - x*A'(x)*A(x) ) = x.
Related series.
Series_Reversion(A(x)) = x - x^2 - 3*x^3 - 22*x^4 - 235*x^5 - 3153*x^6 - 49721*x^7 - 888784*x^8 - 17615520*x^9 + ...
A'(x)*A(x) = x + 3*x^2 + 22*x^3 + 235*x^4 + 3153*x^5 + 49721*x^6 + 888784*x^7 + 17615520*x^8 + ...
A(A(x)) = x + 2*x^2 + 12*x^3 + 110*x^4 + 1294*x^5 + 18127*x^6 + 290620*x^7 + 5206800*x^8 + 102633591*x^9 + ...
A'(A(x)) = 1 + 2*x + 17*x^2 + 208*x^3 + 3108*x^4 + 53328*x^5 + 1018948*x^6 + 21297818*x^7 + 481458997*x^8 + ...
A'(A(x))*A(A(x)) = x + 4*x^2 + 33*x^3 + 376*x^4 + 5242*x^5 + 84625*x^6 + 1534652*x^7 + 30682881*x^8 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A259606, A213591, A303063.

{a(n) = my(A=x); for(i=1, n, A=serreverse(x - x*A'*A +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); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, x^m*(A')^m*A^m/m!)) +O(x^(n+1))); 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)=local(A=x); for(i=1, n, A = x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*(A')^m*A^m/m!)) +O(x^(n+1)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
(1) A( x - x*A'(x)*A(x) ) = x.
(2) A(x) = x + A(x) * A'(A(x)) * A(A(x)).
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * A'(x)^n * A(x)^n / n!.
(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * A'(x)^n * A(x)^n / n! ).
a(n) ~ c * n! * n^alfa / LambertW(1)^n, where alfa = 1.5447806483693... and c = 0.02888888614196289496..., conjecture: alfa = 2*(2*LambertW(1) - 1 + 1/(1 + LambertW(1))). - Vaclav Kotesovec, Feb 22 2023

A213628 G.f. satisfies: A(x) = 1 - (x^2/A(x)) / A( x^2/A(x) ).

Original entry on oeis.org

1, 1, 3, 14, 85, 616, 5072, 46013, 450739, 4702265, 51731956, 595874703, 7147366614, 88905147730, 1143097097833, 15152617826426, 206646826047563, 2894398418226395, 41577147999077079, 611779190051375147, 9211548488261257610, 141802624561414800815

Offset: 1

Paul D. Hanna, Jun 16 2012

nonn

			G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 85*x^5 + 616*x^6 + 5072*x^7 +...
Related expansions:
x^2/A(x) = x - x^2 - 2*x^3 - 9*x^4 - 56*x^5 - 420*x^6 - 3572*x^7 -...
A(x^2/A(x)) = x - x^3 - 7*x^4 - 50*x^5 - 395*x^6 - 3436*x^7 -...
A(x) = x^2/Series_Reversion(G(x)) where G(x) is the g.f. of A213591:
G(x) = x + x^2 + 4*x^3 + 24*x^4 + 178*x^5 + 1512*x^6 + 14152*x^7 +...
such that G(x - G(x)^2) = x.

Cf. A213591, A212411.

{a(n)=local(A=x,G=x); if(n<1, 0, for(i=1, n, G=serreverse(x - G^2+x*O(x^n)));A=x^2/(x-G^2);polcoeff(A, n))}
for(n=1,25,print1(a(n),", "))

G.f.: A(x) = x^2/(x - G(x)^2) where G(x) is the g.f. of A213591 such that G(x^2/A(x)) = G(x - G(x)^2) = x.

G.f.: A(x) = Series_Reversion(x*F(x)) where F(x) = 1 + x*F(1 - 1/F(x))^2 is the g.f. of A212411.

A227852 G.f. A(x) satisfies: A(x) = Series_Reversion( x - (A(x)^2 + A(-x)^2)/2 ).

Original entry on oeis.org

1, 1, 2, 10, 44, 294, 1728, 13389, 93130, 796620, 6235288, 57551130, 493813936, 4857378920, 44989814920, 468103507718, 4633862094852, 50749496457992, 533271010341720, 6126256486912776, 67990630238066888, 817168635245112432, 9541543704324657008, 119719059789052412360

Offset: 1

Paul D. Hanna, Oct 31 2013

nonn

			G.f.: A(x) = x + x^2 + 2*x^3 + 10*x^4 + 44*x^5 + 294*x^6 + 1728*x^7 +...
The series reversion of A(x), G(x) where A(G(x)) = x, begins:
G(x) = x - x^2 - 5*x^4 - 112*x^6 - 4320*x^8 - 227766*x^10 - 14942616*x^12 - 1162657840*x^14 +...+ (-1)^n * A263531(n)*x^(2*n) +...
and can be formed from a bisection of A(x)^2:
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 24*x^5 + 112*x^6 + 716*x^7 + 4320*x^8 + 32290*x^9 + 227766*x^10 + 1893488*x^11 + 14942616*x^12 + 134816212*x^13 + 1162657840*x^14 +...
The related g.f. of A263531, F(x) = -(A(I*x)^2 + A(-I*x)^2)/2, satisfies: F(x) = B(x)^2 - C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1:
F(x) = x^2 - 5*x^4 + 112*x^6 - 4320*x^8 + 227766*x^10 - 14942616*x^12 +...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A263531, A213591, A141202.

{a(n)=local(A=x);for(i=1,n,A=serreverse(x-(A^2+subst(A^2,x,-x +x*O(x^n)))/2));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, Dx(m-1, (A^2+subst(A,x,-x)^2)^m/2^m/m!))+x*O(x^n)); polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))

G.f. A(x) satisfies:
(1) A(x) = x + (A(A(x))^2 + A(-A(x))^2)/2.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) (A(x)^2 + A(-x)^2)^n / (n!*2^n).
(3) (A(I*x)^2 + A(-I*x)^2)/2 = -F(x), where F(x) is the g.f. of A263531 and satisfies: F(x) = B(x)^2 - C(x)^2 such that B(x) + I*C(x) = Series_Reversion(x - I*F(x)), where I^2 = -1.

A276365 G.f. A(x) satisfies: A(x - 2*A(x)^2) = x - A(x)^2.

Original entry on oeis.org

1, 1, 6, 53, 578, 7234, 100044, 1495125, 23802346, 399740086, 7032766196, 128952474242, 2454645604820, 48359400068836, 983683769369624, 20618782389897333, 444636132851851386, 9851377271964349038, 223998085060636514020, 5221799494107885481430, 124695762315403816775932, 3047952254964607540099676, 76206565881709345978097960, 1947752912315470845518308642, 50860833685759573411702643972

Offset: 1

Paul D. Hanna, Sep 01 2016

nonn

			G.f.: A(x) = x + x^2 + 6*x^3 + 53*x^4 + 578*x^5 + 7234*x^6 + 100044*x^7 + 1495125*x^8 + 23802346*x^9 + 399740086*x^10 + 7032766196*x^11 +...
such that A(x - 2*A(x)^2) = x - A(x)^2.
RELATED SERIES.
Note that Series_Reversion(x - 2*A(x)^2) = 2*A(x) - x, which begins:
Series_Reversion(x - 2*A(x)^2) = x + 2*x^2 + 12*x^3 + 106*x^4 + 1156*x^5 + 14468*x^6 + 200088*x^7 + 2990250*x^8 + 47604692*x^9 + 799480172*x^10 +...
Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
R(x) = x - x^2 - 4*x^3 - 28*x^4 - 264*x^5 - 2992*x^6 - 38496*x^7 - 544464*x^8 - 8298080*x^9 - 134500672*x^10 - 2297361024*x^11 +...
then Series_Reversion(x - A(x)^2) = 2*x - R(x), and
R(x) = x - G(x)^2, where G(x) = x + 2*x^2 + 12*x^3 + 108*x^4 + 1208*x^5 + 15536*x^6 + 220832*x^7 + 3390480*x^8 + ... + A177409(n)*x^n + ...
Also, sqrt(A(x) - x) = A(2*A(x) - x), which begins:
sqrt(A(x) - x) = x + 3*x^2 + 22*x^3 + 223*x^4 + 2706*x^5 + 36998*x^6 + 552172*x^7 + 8827263*x^8 + 149328698*x^9 + 2650946274*x^10 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A277295, A213591, A275765, A177409.

m = 26; A[_] = 0;
Do[A[x_] = x + A[2 A[x] - x]^2 + O[x]^m // Normal, {m}];
CoefficientList[A[x]/x, x] (* Jean-François Alcover, Sep 30 2019 *)

{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 - 2*F^2) + F^2, #A) ); A[n]}
for(n=1, 30, print1(a(n), ", "))

G.f. A(x) also satisfies:
(1) A(x) = x + A( 2*A(x) - x )^2.
(2) 2*A(x) = x + Series_Reversion(x - 2*A(x)^2).
(3) R(x) = 2*x - Series_Reversion(x - A(x)^2), where R(A(x)) = x.
(4) R( (x - R(x))^(1/2) ) = 2*x - R(x), where R(A(x)) = x.
(5) A(x) = x + Sum_{n>=1} 2^(n-1) * d^(n-1)/dx^(n-1) A(x)^(2*n) / n!.
(6) A(x) = x + G(A(x))^2, where G(x) = sqrt(x - R(x)) is the g.f. of A177409, and R(A(x)) = x. - Paul D. Hanna, Nov 18 2022
a(n) = Sum_{k=0..n-1} A277295(n,k)*2^k.

A371708 Expansion of g.f. A(x) satisfying A( x*A(x - x^2) ) = x^2.

Original entry on oeis.org

1, 1, 1, 2, 6, 19, 60, 193, 636, 2141, 7331, 25451, 89385, 317036, 1134100, 4087104, 14825482, 54088470, 198348985, 730723956, 2703194553, 10037648254, 37399878530, 139785998185, 523962161491, 1969154471389, 7418488063284, 28010998254007, 105986233046356, 401804972780552

Offset: 1

Paul D. Hanna, Apr 23 2024

nonn

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 60*x^7 + 193*x^8 + 636*x^9 + 2141*x^10 + 7331*x^11 + 25451*x^12 + 89385*x^13 + 317036*x^14 + ...
where A( x*A(x - x^2) ) = x^2.
RELATED SERIES.
Let R(x) be the series reversion of A(x), A(R(x)) = x, which begins
R(x) = x - x^2 + x^3 - 2*x^4 + 2*x^5 - 5*x^6 + 6*x^7 - 16*x^8 + 23*x^9 - 62*x^10 + 100*x^11 - 270*x^12 + 463*x^13 - 1254*x^14 + 2224*x^15 - 6050*x^16 + ...
then R( R(x^2)/x ) = x - x^2.
Also, the bisections B1 and B2 of R(x) are
B1 = (R(x) - R(-x))/2 = x + x^3 + 2*x^5 + 6*x^7 + 23*x^9 + 100*x^11 + 463*x^13 + 2224*x^15 + 10963*x^17 + ...
B2 = (R(x) + R(-x))/2 = -x^2 - 2*x^4 - 5*x^6 - 16*x^8 - 62*x^10 - 270*x^12 - 1254*x^14 - 6050*x^16 + ...
and satisfy B1^2 + B2 = 0 and A(x*B1) = B1^2.
SPECIFIC VALUES.
A( A(1/4) / 2 ) = 1/4 where
A(1/4) = 0.39241307250698647662923990494867613212061604622566765...
A( A(2/9) / 3 ) = 1/9 where
A(2/9) = 0.29957319341272312632777466712131772539171747971866175...
A( A(3/16) / 4 ) = 1/16 where
A(3/16) = 0.2352360051274118086289466324430753987734355106832392...
A( A(4/25) / 5 ) = 1/25 where
A(4/25) = 0.1922953260179964363449115205476634347705922222443464...

Paul D. Hanna, Table of n, a(n) for n = 1..520

Cf. A372528, A272483, A213591, A000108.

{a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( x^2 - subst(Ser(A),x, x*subst(Ser(A),x, x - x^2) ), #A));A[n+1]}
for(n=1,35,print1(a(n),", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n, along with its series reversion R(x), satisfy the following formulas.
(1) A( x*A(x - x^2) ) = x^2.
(2) A(x - x^2) = R(x^2)/x.
(3) (R(x) - R(-x))^2 + 2*(R(x) + R(-x)) = 0.
(4) R(x) = R(-x) - 1 + sqrt(1 - 4*R(-x)).
(5) A(x) = -A( x - 1 + sqrt(1 - 4*x) ).
(6) A(x) = -A(x - 2*C(x)) where C(x) = -C(x - 2*C(x)) is a g.f. of the Catalan numbers (A000108).
(7) A( A(x)*C(x) ) = C(x)^2 where C(x) = (1 - sqrt(1 - 4*x))/2 is a g.f. of the Catalan numbers (A000108).
a(n) ~ c * 4^n / n^(3/2), where c = 0.0517683007874758928168667... - Vaclav Kotesovec, Apr 24 2024

A376176 G.f. A(x) satisfies x = A( x - A(x)^4/x^2 ).

Original entry on oeis.org

1, 1, 6, 55, 622, 8015, 113164, 1711898, 27357970, 457507917, 7952476482, 142972019125, 2648639456048, 50415218306637, 983728646223556, 19641163430509505, 400671660024507294, 8340743906266061866, 176998642509849677206, 3825680705425292568049, 84159282700462688412042

Offset: 1

Paul D. Hanna, Sep 21 2024

nonn

			G.f.: A(x) = x + x^2 + 6*x^3 + 55*x^4 + 622*x^5 + 8015*x^6 + 113164*x^7 + 1711898*x^8 + 27357970*x^9 + 457507917*x^10 + ...
where x = A( x - A(x)^4/x^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 13*x^4 + 122*x^5 + 1390*x^6 + 17934*x^7 + 252847*x^8 + 3814724*x^9 + ...
A(x)^3 = x^3 + 3*x^4 + 21*x^5 + 202*x^6 + 2322*x^7 + 30030*x^8 + 423111*x^9 + 6369930*x^10 + ...
where A(x)^3 = x*A(x)^2 + A(A(x))^4.
A(x)^4 = x^4 + 4*x^5 + 30*x^6 + 296*x^7 + 3437*x^8 + 44600*x^9 + 628454*x^10 + 9446280*x^11 + ...
A(A(x))^4 = x^4 + 8*x^5 + 80*x^6 + 932*x^7 + 12096*x^8 + 170264*x^9 + 2555206*x^10 + 40413484*x^11 + ...
where A(x) = x + A(A(x))^4 / A(x)^2.
A(A(x)) = x + 2*x^2 + 14*x^3 + 141*x^4 + 1712*x^5 + 23392*x^6 + 347444*x^7 + 5498681*x^8 + 91552406*x^9 + ...
A(A(x))^2/A(x) = x + 3*x^2 + 23*x^3 + 242*x^4 + 3017*x^5 + 41965*x^6 + 631381*x^7 + 10089533*x^8 + 169256922*x^9 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A213591, A213639.

{a(n) = my(A=x); if(n<1, 0, for(i=1, n, A=serreverse(x - A^4/x^2 +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, A^(4*m)/x^(2*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, A^(4*m)/x^(2*m+1))/m!)+x*O(x^n))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

b(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(n+j+k, j)/(n+j+k)*b(n-j, 4*j)));
a(n) = b(n-1, 1); \\ Seiichi Manyama, Jun 05 2025

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) x = A( x - A(x)^4/x^2 ).
(2) A(x)^3 = x*A(x)^2 + A(A(x))^4.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(4*n)/x^(2*n) / n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x)^(4*n)/x^(2*n+1) / n! ).
From Seiichi Manyama, Jun 05 2025: (Start)
Let b(n,k) = [x^n] (A(x)/x)^k.
b(n,0) = 0^n; b(n,k) = k * Sum_{j=0..n} binomial(n+j+k,j)/(n+j+k) * b(n-j,4*j).
a(n) = b(n-1,1). (End)

cp's OEIS Frontend

A212411 G.f. satisfies: A(x) = 1 + x*A(1 - 1/A(x))^2.

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A276370 G.f. A(x) satisfies: A( x - A(x) ) = x^2.

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A213639 G.f. A(x) satisfies x = A( x - A(x)^3/x ).

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A277310 G.f. satisfies: A(x - 4*A(x)^2) = x - 3*A(x)^2.

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A360578 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion( x - x*A'(x)*A(x) ).

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A213628 G.f. satisfies: A(x) = 1 - (x^2/A(x)) / A( x^2/A(x) ).

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A227852 G.f. A(x) satisfies: A(x) = Series_Reversion( x - (A(x)^2 + A(-x)^2)/2 ).

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A276365 G.f. A(x) satisfies: A(x - 2*A(x)^2) = x - A(x)^2.

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A277310 G.f. satisfies: A(x - 4A(x)^2) = x - 3A(x)^2.

A360578 Expansion of g.f. A(x) satisfying A(x) = Series_Reversion( x - xA'(x)A(x) ).