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 1-4 of 4 results.

A367390 Expansion of e.g.f. A(x) satisfying A(x)^2 = exp(x) * A(x*A(x)) with A(0) = 0.

Original entry on oeis.org

1, 2, 9, 52, 545, 6366, 98707, 1700840, 35405505, 817958170, 21500633891, 618661892652, 19636408658737, 675144805723766, 25147073628948195, 1004734122294047056, 42965745214637476097, 1955039747566085781426, 94404335950307686644163, 4818562790963397438214100
Offset: 1

Views

Author

Paul D. Hanna, Jan 08 2024

Keywords

Comments

Note that if F(x)^2 = exp(x) * F(x*F(x)) with F(0) = 1, then F(x) is the e.g.f. of A367391.

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 545*x^5/5! + 6366*x^6/6! + 98707*x^7/7! + 1700840*x^8/8! + 35405505*x^9/9! + 817958170*x^10/10! + ...
where A(x)^2 = exp(x) * A(x*A(x)) as can be seen from the following expansions
A(x)^2 = 2*x^2/2! + 12*x^3/3! + 96*x^4/4! + 880*x^5/5! + 11280*x^6/6! + 167664*x^7/7! + 3030944*x^8/8! + ...
A(x*A(x)) = 2*x^2/2! + 6*x^3/3! + 60*x^4/4! + 500*x^5/5! + 7230*x^6/6! + 104202*x^7/7! + 1962296*x^8/8! + ...
Let B(x) = x*A(x), then log( A(x)/x ) equals the sum of all iterations of B(x)
log( A(x)/x ) = x + B(x) + B(B(x)) + B(B(B(x))) + B(B(B(B(x)))) + ...
which is equivalent to
log( A(x)/x ) = x + x*A(x) + x*A(x)*A(x*A(x)) + x*A(x)*A(x*A(x)) * A( x*A(x)*A(x*A(x)) ) + ...
RELATED SERIES.
A(x)/x = 1 + x + 3*x^2/2! + 13*x^3/3! + 109*x^4/4! + 1061*x^5/5! + 14101*x^6/6! + 212605*x^7/7! + 3933945*x^8/8! + 81795817*x^9/9! + ...
log( A(x)/x ) = x + 2*x^2/2! + 6*x^3/3! + 60*x^4/4! + 500*x^5/5! + 6870*x^6/6! + 96642*x^7/7! + 1824536*x^8/8! + 36995688*x^9/9! + ...
Successive iterations of B(x) = x*A(x) begin
B(x) = 2*x^2/2! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 3270*x^6/6! + 44562*x^7/7! + 789656*x^8/8! + ...
B(B(x)) = 24*x^4/4! + 240*x^5/5! + 3600*x^6/6! + 52080*x^7/7! + 994560*x^8/8! + ...
B(B(B(x))) = 40320*x^8/8! + 1451520*x^9/9! + 50803200*x^10/10! + ...
B(B(B(B(x)))) = 20922789888000*x^16/16! + 2845499424768000*x^17/17! + ...
etc.
where A(x) = x * exp(x + B(x) + B(B(x)) + B(B(B(x))) + B(B(B(B(x)))) + ...).

Links

Crossrefs

Cf. A367391, A144681, A367386, A367387.
Cf. A369090.

Programs

  • PARI
    {a(n) = my(A=x, V=[0,1]); for(i=1,n, V = concat(V,0); A = Ser(V);
V[#V] = polcoeff( subst(A,x,x*A) - exp(-x +x*O(x^(#V)))*A^2, #V) ); n!*V[n+1]}
for(n=1,40, print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! and B(x) = x*A(x) satisfies the following formulas.
(1) A(x)^2 = exp(x) * A(x*A(x)).
Let B^n(x) denote the n-th iteration of B(x) = x*A(x), where B^(n+1)(x) = B( B^n(x) ) with B^0(x) = x, then
(2) log( A(x)/x ) = Sum_{n>=0} B^n(x).
(3) B^n(x) = x*A(x)^(2^n - 1) / exp( Sum_{k=0..n-2} (2^(n-k-1) - 1) * B^k(x) ) for n > 1.
(3.a) B^2(x) = x*A(x)^3 / exp(x).
(3.b) B^3(x) = x*A(x)^7 / exp(3*x + B(x)).
(3.c) B^4(x) = x*A(x)^15 / exp(7*x + 3*B(x) + B^2(x)).
(3.d) B^5(x) = x*A(x)^31 / exp(15*x + 7*B(x) + 3*B^2(x) + B^3(x)).
(4) A( B^n(x) ) = A(x)^(2^n) / exp( Sum_{k=0..n-1} 2^(n-k-1) * B^k(x) ) for n > 0.
(4.a) A(B(x)) = A(x)^2 / exp(x).
(4.b) A(B^2(x)) = A(x)^4 / exp(2*x + B(x)).
(4.c) A(B^3(x)) = A(x)^8 / exp(4*x + 2*B(x) + B^2(x)).
(4.d) A(B^4(x)) = A(x)^16 / exp(8*x + 4*B(x) + 2*B^2(x) + B^3(x)).

A369091 Expansion of e.g.f. A(x) satisfying A(x) = x + A( x^2*exp(x) ), with A(0) = 0.

Original entry on oeis.org

1, 2, 6, 36, 260, 2190, 21882, 268856, 3907080, 63977850, 1152946190, 22581979332, 477140664156, 10828556474918, 263163922847490, 6836792356168560, 189694001088036752, 5614994984290505586, 176964200467784915094, 5921022573291003915260, 209568707084236321665060
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! + 6*x^3/3! + 36*x^4/4! + 260*x^5/5! + 2190*x^6/6! + 21882*x^7/7! + 268856*x^8/8! + 3907080*x^9/9! + 63977850*x^10/10! + ...
which equals the sum of all iterations of the function x^2*exp(x).
RELATED SERIES.
x*exp(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! + ... + A369090(n)*x^n/n! + ...
Let R(x) be the series reversion of A(x),
R(x) = x - 2*x^2/2! + 6*x^3/3! - 36*x^4/4! + 340*x^5/5! - 3870*x^6/6! + 52038*x^7/7! - 850472*x^8/8! + 16378920*x^9/9! + ...
then R(x) and e.g.f. A(x) satisfy:
(1) R( A(x) ) = x,
(2) R( A(x) - 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 the sum
A(x) = F(0) + F(1) + F(2) + F(3) + ... + F(n) + ...
equivalently,
A(x) = 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. A369090, A369551 (a(n)/n), A030178.

Programs

  • PARI
    {a(n) = my(A=x); for(i=0, #binary(n),
A = x + subst(A, x, x^2*exp(x +x^2*O(x^n)) )); n! * polcoeff(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) = x + A( x^2*exp(x) ).
(2) A(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) = G(x^2*exp(x))/x is the e.g.f. of A369090.

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 1-4 of 4 results.

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