cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A368630 Expansion of e.g.f. A(x) satisfying A(x/A(x)^2) = exp(x*A(x)).

Original entry on oeis.org

1, 1, 7, 136, 4933, 275536, 21309139, 2137447936, 266227499017, 39924910381312, 7045914488563711, 1437809941831499776, 334581893955246072205, 87792555944973238718464, 25735892905876612366925515, 8363132129019712402301648896, 2992768723058093966270081891089
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2024

Keywords

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.

Examples

			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! + ...

Links

Crossrefs

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

Programs

  • 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),", "))

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)).

A368631 Expansion of e.g.f. A(x) satisfying A(x/A(x)) = exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 7, 118, 3457, 150376, 8869249, 669261160, 62084355505, 6878901271024, 890797404903841, 132568595259161656, 22370325575395442473, 4233795107469842535544, 890606081738110684972705, 206651730919408572588445216, 52550877215770005095599441249, 14564273590596678338725804835680
Offset: 0

Views

Author

Paul D. Hanna, Jan 02 2024

Keywords

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.

Examples

			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! + ...

Links

Crossrefs

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

Programs

  • 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),", "))

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).

A368632 Expansion of e.g.f. A(x) satisfying A(x/A(x)^2) = exp(x*A(x)^2).

Original entry on oeis.org

1, 1, 9, 217, 9521, 634321, 58026745, 6846238057, 998806698209, 174849870369313, 35915074166268521, 8507730512772340345, 2292605150744212481809, 695028316821630097748209, 234883073320203308189545049, 87808334177056337272289692681, 36075481332626610937457504918465
Offset: 0

Views

Author

Paul D. Hanna, Jan 01 2024

Keywords

Examples

			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! + ...

Crossrefs

Cf. A144682, A367385.

Programs

  • 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),", "))

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.
Showing 1-3 of 3 results.

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