A303063 -id:A303063 - cp's OEIS frontend

A360950 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2n) A(x)^n / n!.

1, 2, 12, 108, 1240, 16932, 264740, 4631320, 89270316, 1875586380, 42610756408, 1040307155304, 27157913296228, 754950111249488, 22267948484559720, 694746226969477744, 22863695087986373968, 791675941860401322852, 28776089467457429038620, 1095679176790207081120360

Offset: 0

Paul D. Hanna, Feb 26 2023

nonn

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 108*x^3 + 1240*x^4 + 16932*x^5 + 264740*x^6 + 4631320*x^7 + 89270316*x^8 + 1875586380*x^9 + ...
where
A(x) = 1 + (d/dx x^2*A(x)) + (d^2/dx^2 x^4*A(x)^2)/2! + (d^3/dx^3 x^6*A(x)^3)/3! + (d^4/dx^4 x^8*A(x)^4)/4! + (d^5/dx^5 x^10*A(x)^5)/5! + (d^6/dx^6 x^12*A(x)^6)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^n)/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^2*A(x)), which begins
B(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2822*x^6 + 37820*x^7 + 578915*x^8 + 9918924*x^9 + 187558638*x^10 + ... + A229619(n)*x^n + ...
then A(x) = B'(x) and
B(x) = x * exp( x*A(x) + (d/dx x^3*A(x)^2)/2! + (d^2/dx^2 x^5*A(x)^3)/3! + (d^3/dx^3 x^7*A(x)^4)/4! + (d^4/dx^4 x^9*A(x)^5)/5! + (d^5/dx^5 x^11*A(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^n)/n! + ... ).

Paul D. Hanna, Table of n, a(n) for n = 0..200

Cf. A229619, A356848, A088714, A303063.

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(2*m)*A^m/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^n / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*A(x)).
(3) B(x - x^2*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^n / n! ) is the g.f. of A229619.
(4) a(n) = (n+1) * A229619(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa / LambertW(1)^n, where alfa = 3*LambertW(1) + 1/(1 + LambertW(1)) = 2.33953361459... and c = 0.1926079501120681239... - Vaclav Kotesovec, Feb 27 2023

A360973 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^n / n!.

Original entry on oeis.org

1, 3, 30, 462, 9243, 223014, 6232239, 196780545, 6905085381, 266134485876, 11167349876424, 506653305313320, 24713399794830075, 1289888647516181583, 71744797404818298768, 4237233894492578488668, 264870390088867267319274, 17473793269024825938396135

Offset: 0

Paul D. Hanna, Feb 27 2023

nonn

			G.f.: A(x) = 1 + 3*x^2 + 30*x^4 + 462*x^6 + 9243*x^8 + 223014*x^10 + 6232239*x^12 + 196780545*x^14 + 6905085381*x^16 + ... + a(n)*x^(2*n) + ...
where
A(x) = 1 + (d/dx x^3*A(x)) + (d^2/dx^2 x^6*A(x)^2)/2! + (d^3/dx^3 x^9*A(x)^3)/3! + (d^4/dx^4 x^12*A(x)^4)/4! + (d^5/dx^5 x^15*A(x)^5)/5! + (d^6/dx^6 x^18*A(x)^6)/6! + ... + (d^n/dx^n x^(3*n)*A(x)^n)/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^3*A(x)), which begins
B(x) = x + x^3 + 6*x^5 + 66*x^7 + 1027*x^9 + 20274*x^11 + 479403*x^13 + 13118703*x^15 + 406181493*x^17 + ... + A360976(n)*x^(2*n-1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^2*A(x) + (d/dx x^5*A(x)^2)/2! + (d^2/dx^2 x^8*A(x)^3)/3! + (d^3/dx^3 x^11*A(x)^4)/4! + (d^4/dx^4 x^14*A(x)^5)/5! + (d^5/dx^5 x^17*A(x)^6)/6! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A(x)^n)/n! + ... ).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A360950, A360974, A360975, A360976.

Cf. A229619, A356848, A088714, A303063.

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^m/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A +O(x^(2*n+3))))); polcoeff(A, 2*n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^n / n!.
(2) A(x) = d/dx Series_Reversion(x - x^3*A(x)).
(3) B(x - x^3*A(x)) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^n / n! ) is the g.f. of A360976.
(4) a(n) = (2*n+1) * A360976(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa * 2^n / LambertW(1)^n, where alfa = 1.50465021094584... and c = 0.36053267424501... - Vaclav Kotesovec, Feb 28 2023
alfa = (9*LambertW(1) - 1 + 3/(1 + LambertW(1)))/4. - Vaclav Kotesovec, Mar 13 2023

A360974 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2n) A(x)^(2*n) / n!.

Original entry on oeis.org

1, 2, 18, 260, 4890, 110124, 2844772, 82196424, 2613699450, 90450874860, 3379153837180, 135445714293720, 5796441493971284, 263784018974675416, 12721572505160772840, 648250134428292640272, 34809708051186914034730, 1965040180185473309749788, 116359823755204505172646204

Offset: 0

Paul D. Hanna, Feb 27 2023

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 260*x^3 + 4890*x^4 + 110124*x^5 + 2844772*x^6 + 82196424*x^7 + 2613699450*x^8 + ... + a(n)*x^n + ...
where
A(x) = 1 + (d/dx x^2*A(x)^2) + (d^2/dx^2 x^4*A(x)^4)/2! + (d^3/dx^3 x^6*A(x)^6)/3! + (d^4/dx^4 x^8*A(x)^8)/4! + (d^5/dx^5 x^10*A(x)^10)/5! + (d^6/dx^6 x^12*A(x)^12)/6! + ... + (d^n/dx^n x^(2*n)*A(x)^(2*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^2*A(x)^2), which begins
B(x) = x + x^2 + 6*x^3 + 65*x^4 + 978*x^5 + 18354*x^6 + 406396*x^7 + 10274553*x^8 + 290411050*x^9 + ... + A360977(n)*x^n + ...
then A(x) = B'(x) and
B(x) = x * exp( x*A(x)^2 + (d/dx x^3*A(x)^4)/2! + (d^2/dx^2 x^5*A(x)^6)/3! + (d^3/dx^3 x^7*A(x)^8)/4! + (d^4/dx^4 x^9*A(x)^10)/5! + (d^5/dx^5 x^11*A(x)^12)/6! + ... + (d^(n-1)/dx^(n-1) x^(2*n-1)*A(x)^(2*n))/n! + ... ).

Paul D. Hanna, Table of n, a(n) for n = 0..299

Cf. A360950, A360973, A360975, A360977.

Cf. A229619, A356848, A088714, A303063.

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(2*m)*A^(2*m)/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^2*A^2 +O(x^(n+2))))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(2*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*A(x)^2).
(3) B(x - x^2*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * A(x)^(2*n) / n! ) is the g.f. of A360977.
(4) a(n) = (n+1) * A360977(n+1) for n >= 0.
a(n) ~ c * n! * n^alfa / LambertW(1/2)^n, where alfa = 2.498459235192... and c = 0.0920029178453... - Vaclav Kotesovec, Feb 28 2023
alfa = 5*LambertW(1/2) + 1/(1 + LambertW(1/2)). - Vaclav Kotesovec, Mar 13 2023

A361046 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^(2*n) / n!.

Original entry on oeis.org

1, 3, 45, 1113, 36459, 1448568, 66726309, 3469988835, 200242815669, 12670449226269, 871389659249424, 64693985439491127, 5156607707368927875, 439261264283443326927, 39831856169938193953827, 3831650468281643037364389, 389807188331526942149375433

Offset: 0

Paul D. Hanna, Mar 03 2023

nonn

			G.f.: A(x) = 1 + 3*x^2 + 45*x^4 + 1113*x^6 + 36459*x^8 + 1448568*x^10 + 66726309*x^12 + 3469988835*x^14 + ... + a(n)*x^(2*n) + ...
where
A(x) = 1 + (d/dx x^3*A(x)^2) + (d^2/dx^2 x^6*A(x)^4)/2! + (d^3/dx^3 x^9*A(x)^6)/3! + (d^4/dx^4 x^12*A(x)^8)/4! + (d^5/dx^5 x^15*A(x)^10)/5! + ... + (d^n/dx^n x^(3*n)*A(x)^(2*n))/n! + ...
Related series.
Let B(x) = Series_Reversion(x - x^3*A(x)^2), which begins
B(x) = x + x^3 + 9*x^5 + 159*x^7 + 4051*x^9 + 131688*x^11 + 5132793*x^13 + 231332589*x^15 + 11778989157*x^17 + ... + A361047(n)*x^(2*n-1) + ...
then A(x) = B'(x) and
B(x) = x * exp( x^2*A(x)^2 + (d/dx x^5*A(x)^4)/2! + (d^2/dx^2 x^8*A(x)^6)/3! + (d^3/dx^3 x^11*A(x)^8)/4! + (d^4/dx^4 x^14*A(x)^10)/5! + ... + (d^(n-1)/dx^(n-1) x^(3*n-1)*A(x)^(2*n))/n! + ... ).

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A361047, A360950, A360973, A360974, A360975, A360976.

Cf. A229619, A356848, A088714, A303063.

nt = 40; (* number of terms to produce *)
A[_] = 0;
Do[A[x_] = D[InverseSeries[x - x^3*A[x]^2 + O[x]^k] // Normal, x], {k, 1, 2*nt}];
CoefficientList[A[x^(1/2)], x] (* Jean-François Alcover, Mar 04 2023 *)

{Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); for(i=1, n, A = sum(m=0, n, Dx(m, x^(3*m)*A^(2*m)/m!)) +O(x^(2*n+1))); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))

/* Using series reversion (faster) */
{a(n) = my(A=1); for(i=1, n, A = deriv( serreverse(x - x^3*A^2 +O(x^(2*n+3))))); polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^3*A(x)^2).
(3) B(x - x^3*A(x)^2) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(3*n-1) * A(x)^(2*n) / n! ) is the g.f. of A361047.
(4) a(n) = (2*n+1) * A361047(n+1) for n >= 0.
a(n) == 0 (mod 3) for n > 0.
a(n) ~ c * 2^n * n! * n^alfa / LambertW(1/2)^n, where alfa = 1.623844426394406... and c = 0.18597481905555548924712403113114... - Vaclav Kotesovec, Mar 04 2023
alfa = (15*LambertW(1/2) - 1 + 3/(1 + LambertW(1/2)))/4. - Vaclav Kotesovec, Mar 15 2023

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

A303062 G.f. A(x) satisfies: [x^(n-1)] (1 + x*A(x)^(n-1))^n / A(x)^n = 0 for n>1.

Original entry on oeis.org

1, 1, 2, 8, 60, 643, 8564, 133890, 2376261, 46832442, 1009739331, 23564025488, 590503218735, 15793704933899, 448695132962248, 13487808514722460, 427624923581550100, 14260707306806609885, 499071020445057149835, 18290961984686434723480, 700757535935308305865473, 28017787701624063252219677

Offset: 0

Paul D. Hanna, Apr 17 2018

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 60*x^4 + 643*x^5 + 8564*x^6 + 133890*x^7 + 2376261*x^8 + 46832442*x^9 + 1009739331*x^10 + ...
ILLUSTRATION OF DEFINITION.
The table of coefficients in (1 + x*A(x)^(n-1))^n / A(x)^n begins:
n=1: [1, 0, -2, -6, -50, -565, -7731, -122983, ...];
n=2: [1, 0, -2, -10, -89, -1030, -14307, -230054, ...];
n=3: [1, 0, 0, -9, -111, -1380, -19677, -320958, ...];
n=4: [1, 0, 4, 0, -94, -1520, -23388, -392776, ...];
n=5: [1, 0, 10, 20, 0, -1210, -24030, -436250, ...];
n=6: [1, 0, 18, 54, 225, 0, -18345, -427944, ...];
n=7: [1, 0, 28, 105, 651, 2835, 0, -316344, ...];
n=8: [1, 0, 40, 176, 1364, 8360, 41976, 0, ...]; ...
in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1 + x*A(x)^(n-1))^n / A(x)^n = 0 for n>1.

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A303063, A302702.

{a(n) = my(A=[1]); for(m=1,n+1, A=concat(A,0); A[m] = Vec( (1 + x*Ser(A)^(m-1))^m/Ser(A)^m )[m]/m ); A[n+1]}
for(n=0,30, print1(a(n),", "))

a(n) ~ c * n! * n^(2*LambertW(1)) / LambertW(1)^n, where c = 0.153879081661359639962985708... - Vaclav Kotesovec, Aug 11 2021

A360581 Expansion of A(x) satisfying [x^n] A(x)^n / (1 + x*A(x)^n)^n = 0 for n > 0.

Original entry on oeis.org

1, 1, 3, 17, 131, 1204, 12587, 149131, 2036675, 32358153, 587313706, 11761213199, 252859744189, 5785648936988, 141627609404793, 3737907237793369, 106414467836076985, 3241492594168333618, 104522041356412895455, 3541554178675758259947, 125782730912626755808358

Offset: 0

Paul D. Hanna, Mar 12 2023

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 17*x^3 + 131*x^4 + 1204*x^5 + 12587*x^6 + 149131*x^7 + 2036675*x^8 + 32358153*x^9 + 587313706*x^10 + ...
The table of coefficients in the successive powers of g.f. A(x) begins:
n = 1: [1, 1,  3,  17,  131,  1204,  12587,  149131, ...];
n = 2: [1, 2,  7,  40,  305,  2772,  28657,  335114, ...];
n = 3: [1, 3, 12,  70,  531,  4782,  48936,  565245, ...];
n = 4: [1, 4, 18, 108,  819,  7324,  74272,  848064, ...];
n = 5: [1, 5, 25, 155, 1180, 10501, 105650, 1193530, ...];
n = 6: [1, 6, 33, 212, 1626, 14430, 144208, 1613214, ...];
n = 7: [1, 7, 42, 280, 2170, 19243, 191254, 2120511, ...];
n = 8: [1, 8, 52, 360, 2826, 25088, 248284, 2730872, ...];
...
The table of coefficients in A(x)^n/(1 + x*A(x)^n)^n begins:
n = 1: [1, 0,   2,   12,  100,  955, 10258, 124565, ...];
n = 2: [1, 0,   2,   18,  161, 1606, 17757, 220834, ...];
n = 3: [1, 0,   0,   15,  168, 1806, 21000, 272856, ...];
n = 4: [1, 0,  -4,    0,  114, 1504, 19220, 270692, ...];
n = 5: [1, 0, -10,  -30,    0,  800, 12970, 215445, ...];
n = 6: [1, 0, -18,  -78, -165,    0,  4797, 123990, ...];
n = 7: [1, 0, -28, -147, -364, -329,     0,  32767, ...];
n = 8: [1, 0, -40, -240, -572,  696,  7472,      0, ...];
...
in which the diagonal of all zeros illustrates that
[x^n] A(x)^n / (1 + x*A(x)^n)^n = 0 for n > 0.

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A360582, A360583, A360584, A303063.

{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = -polcoeff( Ser(A)^(#A)/(1 + x*Ser(A)^(#A))^(#A), #A-1)/(#A) );A[n+1]}
for(n=0,30,print1(a(n),", "))

From Vaclav Kotesovec, Mar 13 2023: (Start)
a(n) ~ c * n! * n^alpha / LambertW(1)^n, where alpha = 0.33953... and c = 0.1881608377753...
Conjecture: alpha = 3*LambertW(1) - 2 + 1/(1 + LambertW(1)) = 0.33953361459446... (End)

A365095 Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n-1)xA(x)^2)^n / A(x)^n = 0 for n > 1.

Original entry on oeis.org

1, 1, 4, 27, 256, 3118, 46114, 797049, 15671350, 343712542, 8287906284, 217309849772, 6143454613682, 186012988954448, 5999891924386246, 205262374717093101, 7420869162700453174, 282640364822610119566, 11310634300879858185320, 474456517209788353301282, 20818983374432724237753352

Offset: 0

Paul D. Hanna, Sep 03 2023

nonn

Related identities for the Catalan function C(x) = 1 + x*C(x)^2 (cf. A000108):
(1) [x^(n-1)] (1 + (n-1)*x*C(x))^n / C(x)^n = 0 for n > 1.
(2) [x^(n-1)] (1 + n*x*C(x)^2)^n / C(x)^(2*n) = 0 for n > 1.
(3) [x^(n-1)] (1 + n*x*C(x))^n / C(x)^n = n^(n-1) for n >= 1.
(4) [x^(n-1)] (1 + (n+1)*x*C(x)^2)^n / C(x)^(2*n) = n^(n-1) for n >= 1.

			G.f.: A(x) = 1 + x + 4*x^2 + 27*x^3 + 256*x^4 + 3118*x^5 + 46114*x^6 + 797049*x^7 + 15671350*x^8 + 343712542*x^9 + 8287906284*x^10 + ...
RELATED SERIES.
(1) The power series B(x) = A(x/B(x)) where A(x) = B(x*A(x)) begins
B(x) = 1 + x + 3*x^2 + 17*x^3 + 151*x^4 + 1817*x^5 + 27041*x^6 + 472297*x^7 + 9377293*x^8 + 207254037*x^9 + 5025044843*x^10 + ...
and appears to have only odd coefficients.
(2) The power series D(x) = A(x/D(x)^2) where A(x) = D(x*A(x)^2) begins
D(x) = 1 + x + 2*x^2 + 10*x^3 + 90*x^4 + 1106*x^5 + 16684*x^6 + 293796*x^7 + 5860280*x^8 + 129807560*x^9 + 3149052896*x^10 + ...
modulo 16 of which appears to equal the series
D(x) (mod 16) = (1 + x + 2*x^2 + 10*x^3 + 10*x^4 + 2*x^5 + 12*x^6 + 4*x^7 + 7*x^8 + 7*x^9 - 2*x^10 - 2*x^11 - 10*x^12 + 6*x^13 - 12*x^14 + 4*x^15 - 8*x^16 - 8*x^17) / (1 - x^8).
Explicitly,
D(x) (mod 16) = 1 + x + 2*x^2 + 10*x^3 + 10*x^4 + 2*x^5 + 12*x^6 + 4*x^7 + 8*x^8 + 8*x^9 + 8*x^11 + 8*x^13 + 8*x^15 + 8*x^19 + 8*x^21 + 8*x^23 + ...
RELATED TABLES.
The table of coefficients of x^k in (1 + (n-1)*x*A(x)^2)^n/A(x)^n begins:
n=1: [1, -1,   -3,   -20,  -197,  -2504,  -38396, ...];
n=2: [1,  0,   -4,   -32,  -336,  -4432,  -69620, ...];
n=3: [1,  3,    0,   -41,  -501,  -6795, -107500, ...];
n=4: [1,  8,   24,     0,  -640, -10112, -163272, ...];
n=5: [1, 15,   95,   310,     0, -13027, -246265, ...];
n=6: [1, 24,  252,  1520,  5448,      0, -321580, ...];
n=7: [1, 35,  546,  5033, 30534, 119728,       0, ...]; ...
in which the main diagonal equals all zeros after the initial term, illustrating that [x^(n-1)] (1 + (n-1)*x*A(x)^2)^n / A(x)^n = 0 for n > 1.
The table of coefficients of x^k in A(x)^n begins:
n=1: [1, 1,  4,  27,  256,  3118,  46114, ...];
n=2: [1, 2,  9,  62,  582,  6964, 101241, ...];
n=3: [1, 3, 15, 106,  990, 11667, 166861, ...];
n=4: [1, 4, 22, 160, 1493, 17372, 244658, ...];
n=5: [1, 5, 30, 225, 2105, 24241, 336540, ...];
n=6: [1, 6, 39, 302, 2841, 32454, 444660, ...];
n=7: [1, 7, 49, 392, 3717, 42210, 571438, ...]; ...

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A303063.

{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A;
A[#A] = polcoeff( (1 + (m-1)*x*Ser(A)^2)^m / Ser(A)^m , m-1)/m ); A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) [x^(n-1)] (1 + (n-1)*x*A(x)^2)^n / A(x)^n = 0.
(2) [x^(n-1)] (1 + (k*n-1)*x*A(x)^2)^n / A(x)^n is divisible by n^2 for n > 0 and all integer k (conjecture).

A365516 Expansion of g.f. A(x) satisfying A(x) = (1 + 2xA(x))^2 / (1 + 2xA(x) - 2x^3A(x)^3).

Original entry on oeis.org

1, 2, 4, 10, 32, 112, 400, 1464, 5520, 21296, 83456, 331136, 1328320, 5379200, 21959936, 90271904, 373347840, 1552438016, 6486311680, 27217331456, 114649525760, 484640538112, 2055185596416, 8740711936000, 37273693649920, 159340373710848, 682708771254272, 2931290431277056

Offset: 0

Paul D. Hanna, Sep 07 2023

nonn

Related identities for the Catalan function C(x) = 1 + x*C(x)^2 (A000108):
(1) [x^(n-1)] (1 + n*x*C(x))^n / C(x)^n = n^(n-1) for n >= 1.
(2) [x^(n-1)] (1 + (n+1)*x*C(x)^2)^n / C(x)^(2*n) = n^(n-1) for n >= 1.
Related identity: [x^(n-1)] (1 + (n+1)*x*B(x))^n / B(x)^n = n*(n-1)^(n-1) for n >= 1 when B(x) = 1/(1 - 2*x).
Related identity: F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1), which holds formally for all Maclaurin series F(x). - Paul D. Hanna, Oct 03 2023

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 32*x^4 + 112*x^5 + 400*x^6 + 1464*x^7 + 5520*x^8 + 21296*x^9 + 83456*x^10 + ...
where A(x) = (1 + 2*x*A(x))^2/(1 + 2*x*A(x) - 2*x^3*A(x)^3).
RELATED SERIES.
B(x) = A(x/B(x)) where B(x) = (1 + 2*x)^2/(1 + 2*x - 2*x^3) begins
B(x) = 1 + 2*x + 2*x^3 + 4*x^6 - 8*x^7 + 16*x^8 - 24*x^9 + 32*x^10 - 32*x^11 + 16*x^12 + 32*x^13 - 128*x^14 + ...
RELATED TABLE.
The table of coefficients of x^k in (1 + (n+1)*x*A(x))^n/A(x)^n begins:
n=1: [1,  0,    0,    -2,     -8,    -24,     -76,    -272, ...];
n=2: [1,  2,    1,    -4,    -20,    -64,    -196,    -664, ...];
n=3: [1,  6,   12,     2,    -48,   -192,    -600,   -1896, ...];
n=4: [1, 12,   54,   100,    -23,   -600,   -2224,   -6944, ...];
n=5: [1, 20,  160,   630,   1080,   -696,   -8660,  -30960, ...];
n=6: [1, 30,  375,  2488,   9027,  14406,  -15371, -144852, ...];
n=7: [1, 42,  756,  7546,  44800, 153552,  229376, -342728, ...];
n=8: [1, 56, 1372, 19192, 167222, 921400, 3026332, 4251528, ...]; ...
in which the main diagonal equals n*(n+1)^(n-2) for n > 1.

Paul D. Hanna, Table of n, a(n) for n = 0..400

Cf. A365574, A303063, A365095.

a[n_] := Module[{A = {1}, m, x}, Do[AppendTo[A, 0]; m = Length[A]; A[[-1]] = Coefficient[(1 + (m + 1)*x*SeriesData[x, 0, A, 0, m + 1, 1])^m/SeriesData[x, 0, A, 0, m + 1, 1]^m, x, m - 1]/m - (m + 1)^(m - 2), {i, 1, n}]; A[[n + 1]]]; Table[a[n], {n, 0, 27}] (* Robert P. P. McKone, Sep 07 2023 *)

/* Using series reversion */
{a(n) = my(A = (1/x)*serreverse(x*(1 + 2*x - 2*x^3)/(1 + 2*x +x*O(x^n))^2));
polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

/* Formula [x^(n-1)] (1 + (n+1)*x*A(x))^n/A(x)^n = n*(n+1)^(n-2) */
{a(n) = my(A=[1]); for(i=1,n, A=concat(A,0); m=#A;
A[#A] = polcoeff( (1 + (m+1)*x*Ser(A))^m / Ser(A)^m , m-1)/m - (m+1)^(m-2) );A[n+1]}
for(n=0,30,print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = (1 + 2*x*A(x))^2 / (1 + 2*x*A(x) - 2*x^3*A(x)^3).
(2) A(x/B(x)) = B(x) where B(x) = (1 + 2*x)^2 / (1 + 2*x - 2*x^3).
(3) A(x) = (1/x) * Series_Reversion( x*(1 + 2*x - 2*x^3)/(1 + 2*x)^2 ).
(4) [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+1)^(n-2) for n > 1.
(5) [x^(n-1)] (1 + (n-1)*x*A(x))^n / A(x)^n = -n*(n-3)^(n-2) for n > 1.
(6) [x^(n-1)] (1 + n*x*A(x))^n / A(x)^n = (n^(n-1) - n*(n-2)^(n-2))/2 for n >= 1.
(7) A(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x)^n / (1 + (n+1)*x*A(x))^(n+1). - Paul D. Hanna, Oct 03 2023

A365574 Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n+1)xA(x))^n / A(x)^n = n*(n+2)^(n-2) for n > 1.

Original entry on oeis.org

1, 2, 3, 4, 16, 104, 515, 2090, 8170, 34704, 160014, 751282, 3479758, 16012684, 74362915, 350282602, 1665651094, 7952638460, 38067823370, 182874936368, 882344022104, 4274341269824, 20773195676078, 101228332620524, 494521566769160, 2421729829067636, 11886902458813596

Offset: 0

Paul D. Hanna, Sep 11 2023

nonn

Conjecture 1: a(k) is odd iff k = 2^n - 2 for n >= 1.
Conjecture 2: a(2^n - 2) == 3 (mod 16) for n > 1.
Is there a closed formula for the g.f. of this sequence? Compare to the g.f. of A365516.
Related identities for the Catalan function C(x) = 1 + x*C(x)^2 (A000108):
(1) [x^(n-1)] (1 + n*x*C(x))^n / C(x)^n = n^(n-1) for n >= 1.
(2) [x^(n-1)] (1 + (n+1)*x*C(x)^2)^n / C(x)^(2*n) = n^(n-1) for n >= 1.
Related identity: F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1), which holds formally for all Maclaurin series F(x). - Paul D. Hanna, Oct 03 2023

			G.f.: A(x) = 1 + 2*x + 3*x^2 + 4*x^3 + 16*x^4 + 104*x^5 + 515*x^6 + 2090*x^7 + 8170*x^8 + 34704*x^9 + 160014*x^10 + 751282*x^11 + 3479758*x^12 + ...
RELATED TABLE.
The table of coefficients of x^k in (1 + (n+1)*x*A(x))^n/A(x)^n begins:
n=1: [1,  0,    1,     0,    -11,     -54,    -182,    -594, ...];
n=2: [1,  2,    3,     2,    -21,    -130,    -494,   -1660, ...];
n=3: [1,  6,   15,    20,    -18,    -288,   -1391,   -5070, ...];
n=4: [1, 12,   58,   144,    151,    -468,   -3934,  -17376, ...];
n=5: [1, 20,  165,   720,   1715,    1274,   -8960,  -60530, ...];
n=6: [1, 30,  381,  2650,  10824,   24576,   10623, -176034, ...];
n=7: [1, 42,  763,  7812,  49084,  191016,  413343,   49818, ...];
n=8: [1, 56, 1380, 19600, 176242, 1033664, 3873296, 8000000, ...]; ...
in which the main diagonal equals n*(n+2)^(n-2) for n > 1.

Paul D. Hanna, Table of n, a(n) for n = 0..600

Cf. A365516, A303063, A365095.

/* Formula [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+2)^(n-2) */
{a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); m=#A;
A[#A] = polcoeff( (1 + (m+1)*x*Ser(A))^m / Ser(A)^m , m-1)/m - (m+2)^(m-2) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+2)^(n-2) for n > 1.
(2) [x^(n-1)] (1 + (n-2)*x*A(x))^n / A(x)^n = -2*n*(n-4)^(n-2) for n > 1.
(3) [x^(n-1)] (1 + n*x*A(x))^n / A(x)^n = 2*n*((n+1)^(n-2) - (n-2)^(n-2))/3 for n > 1.
(4) A(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x)^n / (1 + (n+2)*x*A(x))^(n+1). - Paul D. Hanna, Oct 03 2023

cp's OEIS Frontend

A360950 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^n / n!.

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A360973 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^n / n!.

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A360974 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * A(x)^(2*n) / n!.

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A361046 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3*n) * A(x)^(2*n) / n!.

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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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A303062 G.f. A(x) satisfies: [x^(n-1)] (1 + x*A(x)^(n-1))^n / A(x)^n = 0 for n>1.

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A360581 Expansion of A(x) satisfying [x^n] A(x)^n / (1 + x*A(x)^n)^n = 0 for n > 0.

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A365095 Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n-1)*x*A(x)^2)^n / A(x)^n = 0 for n > 1.

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A360950 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2n) A(x)^n / n!.

A360973 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^n / n!.

A360974 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(2n) A(x)^(2*n) / n!.

A361046 Expansion of g.f. A(x) satisfying A(x) = Sum_{n>=0} d^n/dx^n x^(3n) A(x)^(2*n) / n!.

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

A365095 Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n-1)xA(x)^2)^n / A(x)^n = 0 for n > 1.

A365516 Expansion of g.f. A(x) satisfying A(x) = (1 + 2xA(x))^2 / (1 + 2xA(x) - 2x^3A(x)^3).

A365574 Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n+1)xA(x))^n / A(x)^n = n*(n+2)^(n-2) for n > 1.