A144683 -id:A144683

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A144681

E.g.f. satisfies: A(x/A(x)) = exp(x).

+10
6

1, 1, 3, 22, 305, 6656, 204337, 8226436, 414585425, 25315924960, 1828704716801, 153433983789164, 14739472821255481, 1602471473448455104, 195300935112810494801, 26470100501608768436716

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..136

FORMULA

E.g.f. satisfies: A(x) = exp( x*A(log A(x)) ).

E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1).

E.g.f: A(x) = G(2x)^(1/2) where G(x/G(x)^2) = exp(x) and G(x) is the e.g.f. of A144682.

E.g.f: A(x) = G(3x)^(1/3) where G(x/G(x)^3) = exp(x) and G(x) is the e.g.f. of A144683.

E.g.f: A(x) = G(4x)^(1/4) where G(x/G(x)^4) = exp(x) and G(x) is the e.g.f. of A144684.

E.g.f: A(x) = 1/G(-x) where G(x*G(x)) = exp(x) and G(x) is the e.g.f. of A087961.

E.g.f. A(log(A(x))) = log(A(x))/x = G(x) is the e.g.f of A140049 where G(x) satisfies G(x*exp(-x*G(x))) = exp(x*G(x)).

EXAMPLE

E.g.f. A(x) = 1 + x + 3*x^2/2! + 22*x^3/3! + 305*x^4/4! +...

A(x/A(x)) = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! +...

1/A(x) = 1 + x - x^2/2! + 10*x^3/3! - 159*x^4/4! + 3816*x^5/5! -+...

A(log(A(x))) = 1 + x + 5*x^2/2! + 55*x^3/3! + 1005*x^4/4! + 26601*x^5/5! +...

ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(n+1) as follows.

Form a table of coefficients of x^k/k! in exp(x)*A(x)^n for n>=1, k>=0:

exp(x)*A(x)^1: [(1), 2, 6, 35, 416, 8437, 249340, ...];

exp(x)*A(x)^2: [1,(3), 13, 93, 1145, 22593, 645741, ...];

exp(x)*A(x)^3: [1, 4,(22), 181, 2320, 45199, 1257364, ...];

exp(x)*A(x)^4: [1, 5, 33,(305), 4097, 79825, 2177329, ...];

exp(x)*A(x)^5: [1, 6, 46, 471,(6656), 131001, 3529836, ...];

exp(x)*A(x)^6: [1, 7, 61, 685, 10201,(204337), 5477005, ...];

exp(x)*A(x)^7: [1, 8, 78, 953, 14960, 306643,(8226436), ...]; ...

then the terms along the main diagonal form this sequence shift left.

PROG

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(n=0, n, A=exp(serreverse(x/A))); n!*polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^n, n-1))}

CROSSREFS

Cf. A144682, A144683, A144684, A087961, A140049.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 19 2008

STATUS

approved

A144682

E.g.f. satisfies: A(x/A(x)^2) = exp(x).

+10
6

1, 1, 5, 73, 2073, 92481, 5775133, 471058953, 47961475313, 5904863932609, 858738633997941, 144899744540718729, 27970301202134146441, 6106540658691499524993, 1493749158085983126737165

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..14.

FORMULA

E.g.f. satisfies: A(x) = exp( x*A(log A(x))^2 ).

E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2).

E.g.f. A(x) = G(x/2)^2 where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.

EXAMPLE

E.g.f. A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2073*x^4/4! +...

ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2) as follows.

Form a table of coefficients of x^k/k! in exp(x)*A(x)^(2*n) for n>=1, k>=0:

exp(x)*A(x)^2: [(1), 3, 17, 219, 5665, 239283, 14432433, ...];

exp(x)*A(x)^4: [1,(5), 41, 605, 15633, 638325, 37250233, ...];

exp(x)*A(x)^6: [1, 7,(73), 1207, 31825, 1274407, 72322201, ...];

exp(x)*A(x)^8: [1, 9, 113,(2073), 56545, 2249769, 124959057, ...];

exp(x)*A(x)^10:[1, 11, 161, 3251,(92481), 3695451, 202282081, ...];

exp(x)*A(x)^12:[1, 13, 217, 4789, 142705,(5775133), 313637833, ...];

exp(x)*A(x)^14:[1, 15, 281, 6735, 210673, 8688975,(471058953), ...]; ...

then the terms along the main diagonal form this sequence shift left.

PROG

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(n=0, n, A=exp(serreverse(x/A^2))); n!*polcoeff(A, n)}

{a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^(2*n), n-1))}

CROSSREFS

Cf. A144681, A144683, A144684.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 19 2008

STATUS

approved

A144684

E.g.f. satisfies: A(x/A(x)^4) = exp(x).

+10
3

1, 1, 9, 265, 15281, 1379441, 173762425, 28528990393, 5838405067745, 1443332192692321, 421171717332106601, 142520112730201819625, 55149333235223148407953, 24128837930726025305020369

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Table of n, a(n) for n=0..13.

FORMULA

E.g.f. satisfies: A(x) = exp( x*A(log A(x))^4 ).

E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(4*n+4).

E.g.f. A(x) = G(4x)^(1/4) where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.

E.g.f. A(x) = G(2x)^(1/2) where G(x/G(x)^2) = exp(x) and G(x) is the g.f. of A144682.

EXAMPLE

E.g.f. A(x) = 1 + x + 9*x^2/2! + 265*x^3/3! + 15281*x^4/4! +...

ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(4*n+4) as follows.

Form a table of coefficients of x^k/k! in exp(x)*A(x)^(4*n) for n>=1, k>=0:

exp(x)*A(x)^4: [(1), 5, 57, 1565, 84017, 7220725, 879058921, ...];

exp(x)*A(x)^8: [1, (9), 145, 4377, 231713, 19222569, 2265114033, ...];

exp(x)*A(x)^12:[1, 13,(265), 8821, 472273, 38328733, 4391383897, ...];

exp(x)*A(x)^16:[1, 17, 417,(15281), 841025, 67630417, 7578669793, ...];

exp(x)*A(x)^20:[1, 21, 601, 24141, (1379441), 111109701, 12258211401, ...];

exp(x)*A(x)^24:[1, 25, 817, 35785, 2135137, (173762425), 18997491601, ...];

exp(x)*A(x)^28:[1, 29, 1065, 50597, 3161873, 261721069,(28528990393), ...]; ...

then the terms along the main diagonal form this sequence shift left.

PROG

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(n=0, n, A=exp(serreverse(x/A^4))); n!*polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^(4*n), n-1))}

CROSSREFS

Cf. A144681, A144682, A144683.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 19 2008

STATUS

approved

A368630

Expansion of e.g.f. A(x) satisfying A(x/A(x)^2) = exp(x*A(x)).

+10
2

1, 1, 7, 136, 4933, 275536, 21309139, 2137447936, 266227499017, 39924910381312, 7045914488563711, 1437809941831499776, 334581893955246072205, 87792555944973238718464, 25735892905876612366925515, 8363132129019712402301648896, 2992768723058093966270081891089

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Conjecture: a(n) == 1 (mod 3) for n >= 0.

Conjecture: a(2*n) == 1 (mod 2) for n >= 0.

Conjecture: a(2*n+1) == 0 (mod 2) for n >= 1.

LINKS

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

FORMULA

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

(1) A(x/A(x)^2) = exp(x*A(x)).

(2) A(x) = exp(x*B(x)^3) where B(x) = A(x*B(x)^2) = ( (1/x)*Series_Reversion(x/A(x)^2) )^(1/2).

(3) A(x/C(x)^3) = exp(x) where C(x) = A(x/C(x)) = x/Series_Reversion(x*A(x)).

EXAMPLE

E.g.f.: A(x) = 1 + x + 7*x^2/2! + 136*x^3/3! + 4933*x^4/4! + 275536*x^5/5! + 21309139*x^6/6! + 2137447936*x^7/7! + 266227499017*x^8/8! + ...

where A(x/A(x)^2) = exp(x*A(x)) and

exp(x*A(x)) = 1 + x + 3*x^2/2! + 28*x^3/3! + 653*x^4/4! + 28096*x^5/5! + 1833367*x^6/6! + 162874048*x^7/7! + ...

Also,

A(x) = exp(x*B(x)^3) where B(x) = A(x*B(x)^2) begins

B(x) = 1 + x + 11*x^2/2! + 292*x^3/3! + 13149*x^4/4! + 861376*x^5/5! + 75412591*x^6/6! + 8365301568*x^7/7! + ...

B(x)^2 = 1 + 2*x + 24*x^2/2! + 650*x^3/3! + 29360*x^4/4! + 1918482*x^5/5! + 167206144*x^6/6! + ...

B(x)^3 = 1 + 3*x + 39*x^2/2! + 1080*x^3/3! + 49029*x^4/4! + 3199728*x^5/5! + 277840179*x^6/6! + ...

Further,

A(x/C(x)^3) = exp(x) where C(x) = A(x/C(x)) begins

C(x) = 1 + x + 5*x^2/2! + 85*x^3/3! + 2889*x^4/4! + 154441*x^5/5! + 11527693*x^6/6! + 1120674717*x^7/7! + ...

C(x)^3 = 1 + 3*x + 21*x^2/2! + 351*x^3/3! + 11337*x^4/4! + 582843*x^5/5! + 42300765*x^6/6! + ...

PROG

(PARI) {a(n) = my(A=1+x); for(i=0, n, A = exp( x*((1/x)*serreverse( x/(A^2 + x*O(x^n)) ))^(3/2) )); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A368631, A368632, A144682, A144683, A367385.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 02 2024

STATUS

approved

A368631

Expansion of e.g.f. A(x) satisfying A(x/A(x)) = exp(x*A(x)^2).

+10
2

1, 1, 7, 118, 3457, 150376, 8869249, 669261160, 62084355505, 6878901271024, 890797404903841, 132568595259161656, 22370325575395442473, 4233795107469842535544, 890606081738110684972705, 206651730919408572588445216, 52550877215770005095599441249, 14564273590596678338725804835680

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Conjecture: a(n) == 1 (mod 3) for n >= 0.

Conjecture: a(2*n) == 1 (mod 2) for n >= 0.

Conjecture: a(2*n+1) == 0 (mod 2) for n >= 1.

LINKS

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

FORMULA

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

(1) A(x/A(x)) = exp(x*A(x)^2).

(2) A(x) = exp(x*B(x)^3) where B(x) = A(x*B(x)) = (1/x)*Series_Reversion(x/A(x)).

(3) A(x/C(x)^3) = exp(x) where C(x) = A(x/C(x)^2) = ( x/Series_Reversion(x*A(x)^2) )^(1/2).

EXAMPLE

E.g.f.: A(x) = 1 + x + 7*x^2/2! + 118*x^3/3! + 3457*x^4/4! + 150376*x^5/5! + 8869249*x^6/6! + 669261160*x^7/7! + 62084355505*x^8/8! + ...

where A(x/A(x)) = exp(x*A(x)^2) and

exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 61*x^3/3! + 1377*x^4/4! + 49001*x^5/5! + 2476273*x^6/6! + 165555909*x^7/7! + ...

Also,

A(x) = exp(x*B(x)^3) where B(x) = A(x*B(x)) begins

B(x) = 1 + x + 9*x^2/2! + 187*x^3/3! + 6461*x^4/4! + 320721*x^5/5! + 21079255*x^6/6! + 1741882717*x^7/7! + ...

B(x)^3 = 1 + 3*x + 33*x^2/2! + 729*x^3/3! + 25653*x^4/4! + 1275483*x^5/5! + 83368251*x^6/6! + ...

Further,

A(x/C(x)^3) = exp(x) where C(x) = A(x/C(x)^2) begins

C(x) = 1 + x + 3*x^2/2! + 34*x^3/3! + 809*x^4/4! + 30336*x^5/5! + 1584517*x^6/6! + 107443540*x^7/7! + ...

C(x)^2 = 1 + 2*x + 8*x^2/2! + 86*x^3/3! + 1944*x^4/4! + 70802*x^5/5! + 3628996*x^6/6! + ...

PROG

(PARI) {a(n) = my(A=1+x); for(i=0, n, A = exp( x*((1/x)*serreverse( x/(A + x*O(x^n)) ))^3 )); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A368630, A368632, A144682, A144683, A367385.

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 02 2024

STATUS

approved

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