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.

A369090 Expansion of e.g.f. A(x) satisfying A(x) = A( x^2*exp(x) ) / x, with A(0) = 0.

Original entry on oeis.org

1, 2, 9, 52, 425, 4206, 48307, 632360, 9444465, 159240250, 2983729331, 61300668012, 1367054727337, 32844312889766, 845234187028155, 23190947446000336, 675895337644401377, 20863665943202969586, 680448552777544884643, 23395823324931227353940, 846248620848062865320601
Offset: 1

Views

Author

Paul D. Hanna, Jan 26 2024

Keywords

Comments

Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 425*x^5/5! + 4206*x^6/6! + 48307*x^7/7! + 632360*x^8/8! + 9444465*x^9/9! + 159240250*x^10/10! + ...
RELATED SERIES.
The expansion of the logarithm of A(x)/x starts
log(A(x)/x) = x + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2190*x^6/6! + 21882*x^7/7! + 268856*x^8/8! + ... + A369091(n)*x^n/n! + ...
and equals the sum of all iterations of the function x^2*exp(x).
Let R(x) be the series reversion of A(x),
R(x) = x - 2*x^2/2! + 3*x^3/3! + 8*x^4/4! - 155*x^5/5! + 1464*x^6/6! - 7931*x^7/7! - 65360*x^8/8! + 2742345*x^9/9! + ...
then R(x) and e.g.f. A(x) satisfy:
(1) R( A(x) ) = x,
(2) R( x*A(x) ) = x^2 * exp(x).
GENERATING METHOD.
Let F(n) equal the n-th iteration of x^2*exp(x), so that
F(0) = x,
F(1) = x^2 * exp(x),
F(2) = x^4 * exp(2*x) * exp(x^2*exp(x)),
F(3) = x^8 * exp(4*x) * exp(2*x^2*exp(x)) * exp(F(2)),
F(4) = x^16 * exp(8*x) * exp(4*x^2*exp(x)) * exp(2*F(2)) * exp(F(3)),
F(5) = x^32 * exp(16*x) * exp(8*x^2*exp(x)) * exp(4*F(2)) * exp(2*F(3)) * exp(F(4)),
...
F(n+1) = F(n)^2 * exp(F(n))
...
Then the e.g.f. A(x) equals
A(x) = x * exp(F(0) + F(1) + F(2) + F(3) + ... + F(n) + ...).
equivalently,
A(x) = x * exp(x + x^2*exp(x) + x^4*exp(2*x)*exp(x^2*exp(x)) + x^8*exp(4*x)*exp(2*x^2*exp(x)) * exp(x^4*exp(2*x)*exp(x^2*exp(x))) + ...).

Links

Crossrefs

Cf. A369091, A369550 (a(n)/n), A030178.
Cf. A367390.

Programs

  • PARI
    {a(n) = my(A=x); for(i=0, #binary(n),
A = subst(A, x, x^2*exp(x +x^2*O(x^n)) )/x ); n! * polcoeff(H=A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = A(x^2*exp(x)) / x.
(2) R(x*A(x)) = x^2*exp(x), where R(A(x)) = x.
(3) A(x) = x * exp( Sum_{n>=0} F(n) ), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
(4) A(x) = x * exp(L(x)), where L(x) = x + L(x^2*exp(x)) is the e.g.f. of A369091.

A369550 Expansion of e.g.f. A(x) satisfying A(x) = exp(x) * A(x^2*exp(x)).

Original entry on oeis.org

1, 1, 3, 13, 85, 701, 6901, 79045, 1049385, 15924025, 271248121, 5108389001, 105158055949, 2346022349269, 56348945801877, 1449434215375021, 39758549273200081, 1159092552400164977, 35813081725133941297, 1169791166246561367697, 40297553373717279300981, 1460613225168596836153741
Offset: 0

Views

Author

Paul D. Hanna, Jan 29 2024

Keywords

Comments

Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! + 6901*x^6/6! + 79045*x^7/7! + 1049385*x^8/8! + 15924025*x^9/9! + ...
RELATED SERIES.
The expansion of A(x^2*exp(x)) begins
exp(-x) * A(x) = A(x^2*exp(x)) = 1 + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 380*x^5/5! + 3750*x^6/6! +  + 42882*x^7/7! + 576296*x^8/8! + ...
The logarithm of e.g.f. A(x) equals L(x) where L(x) = x + L(x^2*exp(x)),
L(x) = x + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2190*x^6/6! + 21882*x^7/7! + 268856*x^8/8! + ... + A369091(n)*x^n/n! + ...

Crossrefs

Cf. A369090, A369091, A369551.

Programs

  • PARI
    {a(n) = my(A=1+x, X = x + x*O(x^n)); for(i=1,n, A = exp(X) * subst(A,x,x^2*exp(X)) ); n!*polcoeff(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = exp(x) * A(x^2*exp(x)).
(2) A(x) = exp( Sum_{n>=0} F(n) ), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
(3) A(x) = exp(L(x)) where L(x) = x + L(x^2*exp(x)) is the e.g.f of A369091.
(4) A(x) = G(x)/x where G(x) = G(x^2*exp(x))/x is the e.g.f. of A369090.
a(n) = A369090(n+1)/(n+1) for n >= 0.

A369551 Expansion of e.g.f. A(x) satisfying A(x) = 1 + x*exp(x) * A(x^2*exp(x)).

Original entry on oeis.org

1, 1, 2, 9, 52, 365, 3126, 33607, 434120, 6397785, 104813290, 1881831611, 36703128012, 773468319637, 17544261523166, 427299522260535, 11158470652237456, 311944165793916977, 9313905287778153426, 296051128664550195763, 9979462242106491507860, 355292353569342771519021
Offset: 0

Views

Author

Paul D. Hanna, Jan 29 2024

Keywords

Comments

Limit (a(n)/n!)^(1/n) = 1/w where w*exp(w) = 1 and w = LambertW(1) = 0.567143290409783872999968... (cf. A030178).

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 365*x^5/5! + 3126*x^6/6! + 33607*x^7/7! + 434120*x^8/8! + 6397785*x^9/9! + 104813290*x^10/10! + ...
RELATED SERIES.
The expansion of A(x^2*exp(x)) begins
A(x^2*exp(x)) = 1 + 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2550*x^6/6! + 29442*x^7/7! + 386456*x^8/8! + ...
where A(x) = 1 + x*exp(x) * A(x^2*exp(x)).
The expansion of exp(x*A(x)) is the e.g.f. of A369550, which begins
exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! + 6901*x^6/6! + 79045*x^7/7! + 1049385*x^8/8! + ... + A369550(n)*x^n/n! + ...

Crossrefs

Cf. A369090, A369091, A369550.

Programs

  • PARI
    {a(n) = my(A=1+x, X = x + x*O(x^n)); for(i=1, n, A = 1 + x*exp(X) * subst(A, x, x^2*exp(X)) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + x*exp(x) * A(x^2*exp(x)).
(2) A(x) = (1/x) * Sum_{n>=0} F(n), where F(0) = x, and F(n+1) = F(n)^2 * exp(F(n)) for n >= 0.
(3) A(x) = log(G(x)) / x where G(x) = exp(x) * G(x^2*exp(x)) is the e.g.f. of A369550.
(4) A(x) = L(x)/x where L(x) = x + L(x^2*exp(x)) is the e.g.f of A369091.
a(n) = A369091(n+1)/(n+1) for n >= 0.
Showing 1-3 of 3 results.

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