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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
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))}
E.g.f. satisfies: A(x/A(x)^3) = exp(x).
+10
5
1, 1, 7, 154, 6625, 446776, 42088609, 5171653432, 792466370161, 146737621112464, 32079442602647521, 8134165156479090136, 2358873194743497162889, 773523517692799444058632, 284154419348084944647780289
FORMULA
E.g.f. satisfies: A(x) = exp( x*A(log A(x))^3 ).
E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(3*n+3).
E.g.f. A(x) = G(3x)^(1/3) where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.
EXAMPLE
E.g.f. A(x) = 1 + x + 7*x^2/2! + 154*x^3/3! + 6625*x^4/4! +...
ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(3*n+3) as follows.
Form a table of coefficients of x^k/k! in exp(x)*A(x)^(3*n) for n>=1, k>=0:
exp(x)*A(x)^3: [(1), 4, 34, 685, 27256, 1747159, 159049000, ...];
exp(x)*A(x)^6: [1, (7), 85, 1909, 75193, 4654417, 410053357, ...];
exp(x)*A(x)^9: [1, 10,(154), 3835, 153208, 9284725, 795356632, ...];
exp(x)*A(x)^12:[1, 13, 241, (6625), 272641, 16385713, 1373165425, ...];
exp(x)*A(x)^15:[1, 16, 346, 10441,(446776), 26918851, 2221660936, ...];
exp(x)*A(x)^18:[1, 19, 469, 15445, 690841, (42088609), 3443635405, ...];
exp(x)*A(x)^21:[1, 22, 610, 21799, 1022008, 63371617,(5171653432), ...]; ...
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^3))); 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^(3*n), n-1))}
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
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))}
Expansion of e.g.f. A(x) satisfying A(x/A(x)^2) = exp(x*A(x)).
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2
1, 1, 7, 136, 4933, 275536, 21309139, 2137447936, 266227499017, 39924910381312, 7045914488563711, 1437809941831499776, 334581893955246072205, 87792555944973238718464, 25735892905876612366925515, 8363132129019712402301648896, 2992768723058093966270081891089
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.
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), ", "))
Expansion of e.g.f. A(x) satisfying A(x/A(x)) = exp(x*A(x)^2).
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2
1, 1, 7, 118, 3457, 150376, 8869249, 669261160, 62084355505, 6878901271024, 890797404903841, 132568595259161656, 22370325575395442473, 4233795107469842535544, 890606081738110684972705, 206651730919408572588445216, 52550877215770005095599441249, 14564273590596678338725804835680
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.
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), ", "))
Expansion of e.g.f. A(x) satisfying A(x/A(x)^2) = exp(x*A(x)^2).
+10
2
1, 1, 9, 217, 9521, 634321, 58026745, 6846238057, 998806698209, 174849870369313, 35915074166268521, 8507730512772340345, 2292605150744212481809, 695028316821630097748209, 234883073320203308189545049, 87808334177056337272289692681, 36075481332626610937457504918465
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).
(2) A(x) = exp(x*B(x)^4) where B(x) = A(x*B(x)^2) = ( (1/x)*Series_Reversion(x/A(x)^2) )^(1/2).
(3) A(x/C(x)^4) = exp(x) where C(x) = A(x/C(x)^2) = ( x/Series_Reversion(x*A(x)^2) )^(1/2).
(4) A(x)^2 = F(2*x) where F(x/F(x)) = exp(x*F(x)) and F(x) is the e.g.f. of A367385.
EXAMPLE
E.g.f.: A(x) = 1 + x + 9*x^2/2! + 217*x^3/3! + 9521*x^4/4! + 634321*x^5/5! + 58026745*x^6/6! + 6846238057*x^7/7! + 998806698209*x^8/8! + ...
where A(x/A(x)^2) = exp(x*A(x)^2) and
exp(x*A(x)^2) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2265*x^4/4! + 119361*x^5/5! + 9255133*x^6/6! + 965731593*x^7/7! + ...
A(x)^2 = 1 + 2*x + 20*x^2/2! + 488*x^3/3! + 21264*x^4/4! + 1402912*x^5/5! + 127177792*x^6/6! + 14889247872*x^7/7! + ...
Also,
A(x) = exp(x*B(x)^4) where B(x) = A(x*B(x)^2) begins
B(x) = 1 + x + 13*x^2/2! + 409*x^3/3! + 21769*x^4/4! + 1680161*x^5/5! + 172774357*x^6/6! + 22446379705*x^7/7! + ...
B(x)^2 = 1 + 2*x + 28*x^2/2! + 896*x^3/3! + 47824*x^4/4! + 3684352*x^5/5! + 377546176*x^6/6! + ...
B(x)^4 = 1 + 4*x + 64*x^2/2! + 2128*x^3/3! + 114688*x^4/4! + 8826944*x^5/5! + 899745280*x^6/6! + ...
Further,
A(x/C(x)^4) = exp(x) where C(x) = A(x/C(x)^2) begins
C(x) = 1 + x + 5*x^2/2! + 97*x^3/3! + 3801*x^4/4! + 233681*x^5/5! + 20005213*x^6/6! + 2225362161*x^7/7! + ...
C(x)^2 = 1 + 2*x + 12*x^2/2! + 224*x^3/3! + 8528*x^4/4! + 515072*x^5/5! + 43572928*x^6/6! + ...
C(x)^4 = 1 + 4*x + 32*x^2/2! + 592*x^3/3! + 21504*x^4/4! + 1254464*x^5/5! + 103581184*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)) ))^2 )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
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