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.

A163138 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2^n + A(x))^n * x^n/n ).

Original entry on oeis.org

1, 3, 20, 329, 22584, 7938470, 12605643936, 84977963809781, 2379247465188706528, 273419351336298753589802, 128009562526607810326874017088, 242979581192696030760182903464959706
Offset: 0

Views

Author

Paul D. Hanna, Aug 07 2009

Keywords

Comments

More generally, we have the following identity:
If A(x,q) = exp( Sum_{n>=1} (q^n + A(x,q))^n * x^n/n ), then
A(x,q) = 1/(1-x*A(x,q))*exp( Sum_{n>=1} q^(n^2)/(1-q^n*x*A(x,q))^n*x^n/n ).
Conjecture: if q is an integer, then A(x,q) is a power series in x with integer coefficients.
Setting q=1 defines the g.f. of the large Schroeder numbers (A006318).

Examples

			G.f.: A(x) = 1 + 3*x + 20*x^2 + 329*x^3 + 22584*x^4 + 7938470*x^5 +...
log(A(x)) = [2 + A(x)]*x + [2^2 + A(x)]^2*x^2/2 + [2^3 + A(x)]^3*x^3/3 +...
log(A(x)*(1-xA(x))) = 2/(1-2xA(x))*x + 2^4/(1-4xA(x))^2*x^2/2 + 2^9/(1-8xA(x))^3*x^3/3 +...
log(A(x)) = 3*x + 31*x^2/2 + 834*x^3/3 + 86227*x^4/4 + 39339038*x^5/5 +...

Crossrefs

Cf. A202518, A155200.

Programs

  • Mathematica
    m = 12; A[] = 1; Do[A[x] = Exp[Sum[(2^n + A[x])^n x^n/n, {n, 1, m}]] + O[x]^m, {m}]; CoefficientList[A[x], x] (* Jean-François Alcover, Nov 03 2019 *)
  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (2^m+A+x*O(x^n))^m*x^m/m))); polcoeff(A, n)}

Formula

G.f.: A(x) = 1/(1-x*A(x))*exp( Sum_{n>=1} 2^(n^2)/(1 - 2^n*x*A(x))^n * x^n/n ).

Extensions

Comment corrected by Paul D. Hanna, Aug 08 2009

A202668 G.f. satisfies: A(x) = exp( Sum_{n>=1} (A(x) - (-1)^n)^n * x^n/n ).

Original entry on oeis.org

1, 2, 4, 12, 42, 158, 618, 2498, 10360, 43832, 188420, 820608, 3613212, 16057640, 71933768, 324482500, 1472604586, 6719100254, 30804229858, 141829955338, 655541387406, 3040527731790, 14147444737654, 66018910398574, 308898542610666, 1448867831911170
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2011

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 12*x^3 + 42*x^4 + 158*x^5 + 618*x^6 + ...
where
log(A(x)) = (A(x) + 1)*x + (A(x) - 1)^2*x^2/2 + (A(x) + 1)^3*x^3/3 + (A(x) - 1)^4*x^4/4 + ...
log( A(x)*(1-x*A(x)) ) = 1/(1 + x*A(x))*x + 1/(1 - x*A(x))^2*x^2/2 + 1/(1 + x*A(x))^3*x^3/3 + 1/(1 - x*A(x))^4*x^4/4 + ...
From _Paul D. Hanna_, Oct 11 2024: (Start)
SPECIFIC VALUES.
A(t) = 2 at t = 0.195782060076367892865630673522992184838101...
where 12*t^3 - 4*t^2 - 15*t + 3 = 0.
A(t) = 3/2 at t = 0.1528468026979892250300352740045422934687...
where 45*t^3 - 18*t^2 - 260*t + 40 = 0.
A(1/6) = 1.5975588141693553913621853542774164447766461118908...
A(1/7) = 1.4422077780342017637064340698606478883307441400444...
A(1/8) = 1.3558965312086216338851741626422486193364696459775...
A(1/9) = 1.2992876417963412242026519185070094965390617289384...
A(1/10) = 1.258828814568496961617240364573696812116531654741...
(End)

Links

Crossrefs

Cf. A202669, A185385, A202630, A202518, A155200.

Programs

  • PARI
    {a(n) = my(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (A - (-1)^m +x*O(x^n))^m * x^m/m))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

Formula

G.f. satisfies: A(x) = 1/(1-x*A(x)) * exp( Sum_{n>=1} 1/(1 - (-1)^n*x*A(x))^n * x^n/n ).
G.f. satisfies: A(x) = sqrt( (1 - (A(x)+1)^2*x^2)/(1 - (A(x)-1)^2*x^2) ) / (1 - (A(x)+1)*x).
G.f. satisfies: 0 = -(1+x) - x*A(x) + (1+x)*(1-x)^2*A(x)^2 - x*(1-x)^2*A(x)^3 - x^2*(1+x)*A(x)^4 + x^3*A(x)^5.
From Vaclav Kotesovec, Oct 11 2024: (Start)
a(n) ~ sqrt((-1 - s + (-1 - 2*r + 3*r^2)*s^2 + (-1 + 4*r - 3*r^2)*s^3 - r*(2 + 3*r)*s^4 + 3*r^2*s^5)/(1 - r*(1 + 3*s) + r^2*(-1 + 6*s - 6*s^2) + r^3*(1 - 3*s - 6*s^2 + 10*s^3))) / (2*sqrt(Pi) * n^(3/2) * r^(n - 1/2)), where r = 0.20089689587759865228481815120918189691453519374477284069915... and s = 2.3487742728380350386577466365052703249852809669846393564277... are positive real roots of the system of equations s^2*(1 + r^3*(-1 + s)^2*(1 + s)) = 1 + r^2*(-1 + s)^2*s^2 + r*(1 + s + s^2 + s^3) and 2*(-1 + r)^2*(1 + r)*s + 5*r^3*s^4 = r*(1 + 3*(-1 + r)^2*s^2 + 4*r*(1 + r)*s^3).
Numerically, a(n) ~ c * d^n / n^(3/2), where d = 1/r = 4.977677706923229216140896605827075562322447814212438341196056039... and c = 0.7100736662419384614471705442776864037581200760804364785319... (End)

A185385 G.f. satisfies: A(x) = exp( Sum_{n>=1} (2*A(x) - (-1)^n)^n * x^n/n ).

Original entry on oeis.org

1, 3, 11, 61, 381, 2527, 17559, 126265, 931321, 7007035, 53568131, 414929621, 3249392917, 25684315319, 204645707183, 1641910625009, 13253684541553, 107561523423731, 877109999610107, 7183095973808493, 59053492869471661, 487189276030904207, 4032100262853037127
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2011

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 11*x^2 + 61*x^3 + 381*x^4 + 2527*x^5 + 17559*x^6 +...
where
log(A(x)) = (2*A(x) + 1)*x + (2*A(x) - 1)^2*x^2/2 + (2*A(x) + 1)^3*x^3/3 + (2*A(x) - 1)^4*x^4/4 +...
log(A(x)*(1-2*x*A(x))) = 1/(1 + 2*x*A(x))*x + 1/(1 - 2*x*A(x))^2*x^2/2 + 1/(1 + 2*x*A(x))^3*x^3/3 + 1/(1 - 2*x*A(x))^4*x^4/4 +...

Crossrefs

Cf. A202519, A163138, A202668, A202630, A202518, A155200.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (2*A-(-1)^m+x*O(x^n))^m*x^m/m))); polcoeff(A, n)}

Formula

G.f. satisfies: A(x) = 1/(1-2*x*A(x)) * exp( Sum_{n>=1} 1/(1 - (-1)^n*2*x*A(x))^n * x^n/n ).
G.f. satisfies: A(x) = sqrt( (1 - (2*A(x)+1)^2*x^2)/(1 - (2*A(x)-1)^2*x^2) ) / (1 - (2*A(x)+1)*x).
G.f. satisfies: 0 = -(1+x) - 2*x*A(x) + (1+x)*(1-x)^2*A(x)^2 - 2*x*(1-x)^2*A(x)^3 - 2^2*x^2*(1+x)*A(x)^4 + 2^3*x^3*A(x)^5.

A202629 G.f.: exp( Sum_{n>=1} (3^n - A(x))^n * x^n/n ).

Original entry on oeis.org

1, 2, 32, 5872, 10244654, 166008832278, 24810745551644598, 34076373857728228215714, 428687442859626139066325301140, 49247086410581981443124673896698437124, 51529024823944797258322973430879108808780359272
Offset: 0

Views

Author

Paul D. Hanna, Dec 21 2011

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 32*x^2 + 5872*x^3 + 10244654*x^4 + 166008832278*x^5 +...
where
log(A(x)) = (3 - A(x))*x + (3^2 - A(x))^2*x^2/2 + (3^3 - A(x))^3*x^3/3 + (3^4 - A(x))^4*x^4/4 +...

Crossrefs

Cf. A202630, A202518, A155203.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,(3^m-A+x*O(x^n))^m*x^m/m)));polcoeff(A,n)}
Showing 1-4 of 4 results.

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