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.

A370440 Expansion of g.f. A(x) satisfying A(x) = A( x^3 + 3*x^2*A(x)^2 )^(1/3), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 15, 30, 55, 113, 274, 683, 1596, 3547, 7990, 18968, 46530, 113663, 273392, 656421, 1598270, 3951520, 9827565, 24411649, 60599823, 150978177, 378293690, 951828992, 2398983638, 6051008950, 15284145261, 38690832455, 98154905623, 249390491237, 634296702273
Offset: 1

Views

Author

Paul D. Hanna, Mar 09 2024

Keywords

Comments

Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 15*x^7 + 30*x^8 + 55*x^9 + 113*x^10 + 274*x^11 + 683*x^12 + 1596*x^13 + 3547*x^14 + 7990*x^15 + ...
where A(x)^3 = A( x^3 + 3*x^2*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 18*x^7 + 47*x^8 + 106*x^9 + 216*x^10 + 450*x^11 + 1040*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 18*x^7 + 42*x^8 + 109*x^9 + 264*x^10 + 585*x^11 + 1270*x^12 + ...
Let B(x) denote the series reversion of A(x), A(B(x)) = x,
B(x) = x - x^2 + x^3 - x^4 + x^6 - x^7 + 2*x^9 - 3*x^10 + 6*x^12 - 9*x^13 + 20*x^15 - 30*x^16 + 71*x^18 - 110*x^19 + 267*x^21 - 419*x^22 + 1041*x^24 - 1648*x^25 + 4168*x^27 - 6652*x^28 + 17047*x^30 + ...
then B(x^3) = B(x)^3 + 3*x^2*B(x)^2, where
B(x)^2 = x^2 - 2*x^3 + 3*x^4 - 4*x^5 + 3*x^6 - 3*x^8 + 4*x^9 - 8*x^11 + 11*x^12 - 23*x^14 + 34*x^15 + ...
B(x)^3 = x^3 - 3*x^4 + 6*x^5 - 10*x^6 + 12*x^7 - 9*x^8 + x^9 + 9*x^10 - 12*x^11 - x^12 + 24*x^13 - 33*x^14 + 69*x^16 - 102*x^17 + ...
Further, the trisections of B(x) = C1(x) + C2(x) + C3(x) are
C1(x) = x^4/C3(x) = x - x^4 - x^7 - 3*x^10 - 9*x^13 - 30*x^16 - 110*x^19 - ...
C2(x) = -x^2, and
C3(x) = x^3 + x^6 + 2*x^9 + 6*x^12 + 20*x^15 + 71*x^18 + 267*x^21 + 1041*x^24 + 4168*x^27 + 17047*x^30 + 70902*x^33 + ... + A370446(n)*x^(3*n) + ...
Compare these series to the series trisections involved in series reversion of A264228.
SPECIFIC VALUES.
A(1/3) = 0.5339969110985873619406256103732700685272...
A(1/4) = 0.3373018860609501862067597141160425025580...
A(1/5) = 0.2509433336474255853462277222741392614966...
A(1/6) = 0.2003115176013404351183299069966738623357...
A(1/8) = 0.1429156905534693639298206599148805278651...
A(1/3)^3 = A(1/27 + 3*A(1/3)^2/9) = A(0.132087937391...) = 0.152270661558...
A(1/4)^3 = A(1/64 + 3*A(1/4)^2/16) = A(0.036957355438...) = 0.038375699859...
A(1/5)^3 = A(1/125 + 3*A(1/5)^2/25) = A(0.015556706804...) = 0.250943333647...

Links

Crossrefs

Cf. A370441, A370446, A264228, A356781.

Programs

  • PARI
    {a(n) = my(A=[1],G); for(i=1,n, G = x*Ser(A); A = Vec((subst(G,x, x^3 + 3*x^2*G^2) + x^4*O(x^#A))^(1/3)); );A[n+1]}
for(n=0,40, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n) * x^n satisfies the following formulas.
(1) A(x) = A( x^3 + 3*x^2*A(x)^2 )^(1/3).
(2) B(x^3) = B(x)^3 + 3*x^2*B(x)^2, where A(B(x)) = x.
a(n) ~ c * d^n / n^(3/2), where d = 2.653503750287... and c = 0.193303... - Vaclav Kotesovec, Mar 14 2024

A370540 Expansion of g.f. A(x) satisfying A(x)^2 = A(x^2) * (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

Original entry on oeis.org

1, 1, 4, 12, 45, 157, 584, 2155, 8110, 30587, 116326, 443984, 1702272, 6546563, 25252094, 97638658, 378351696, 1468876958, 5712276601, 22247635905, 86765271643, 338795469496, 1324374411164, 5182303804184, 20297243177269, 79564763550396, 312137086267106, 1225421059470049
Offset: 0

Views

Author

Paul D. Hanna, Mar 12 2024

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 12*x^3 + 45*x^4 + 157*x^5 + 584*x^6 + 2155*x^7 + 8110*x^8 + 30587*x^9 + 116326*x^10 + 443984*x^11 + ...
RELATED SERIES.
We may illustrate the formulas using the following related series expansions.
Recall that the Catalan function C(x) = (1 - sqrt(1-4*x))/2 begins
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + ... + A000108(n)*x^n + ...
(1) By definition, A(x) = sqrt( A(x^2) * F(x) ) where
F(x) = (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) begins
F(x) = 1 + 2*x + 8*x^2 + 30*x^3 + 118*x^4 + 462*x^5 + 1824*x^6 + 7208*x^7 + 28558*x^8 + ... + A370539(n)*x^n + ...
(2) Also, G(x) = G( x^2 + 2*x^2*G(x) )^(1/2) begins
G(x) = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 14*x^7 + 32*x^8 + 74*x^9 + 172*x^10 + 408*x^11 + ... + A356781(n)*x^n + ...
such that the series reversion of G(x) equals
x*A(x^2)*(1 - x*C(x^2)) = x - x^2 + x^3 - 2*x^4 + 4*x^5 - 7*x^6 + 12*x^7 - 23*x^8 + 45*x^9 - 84*x^10 + 157*x^11 - 302*x^12 + 584*x^13 - 1121*x^14 + ...

Links

Crossrefs

Cf. A370539, A356781, A000108.

Programs

  • PARI
    {a(n) = my(x = 'x + O('x^(n+4)), C(x) = (1 - sqrt(1 - 4*x))/(2*x), A = 1+x); for(i=1,n, A = sqrt( subst(A,'x,x^2) * (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) ) ); polcoeff(A,n);}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n and C(x) = (1 - sqrt(1-4*x))/2 satisfy the following formulas.
(1) A(x)^2 = A(x^2) * F(x) where F(x) = (1 - x*C(x)) * (1 - x*C(x^2)) / (1 - 4*x) is the g.f. of A370539.
(2) G( x*A(x^2)*(1 - x*C(x^2)) ) = x, where G(x) = G( x^2 + 2*x^2*G(x) )^(1/2) is the g.f. of A356781.
a(n) ~ c * 4^n / sqrt(n), where c = 0.3550434768046000612979284344613941075803... - Vaclav Kotesovec, Mar 14 2024

A356780 Coefficients in the odd function A(x) such that: A(x) = A( x^2 + 2*x^2*A(x)^2 )^(1/2), with A(0)=0, A'(0)=1.

Original entry on oeis.org

1, 1, 2, 6, 21, 78, 303, 1223, 5085, 21623, 93585, 410894, 1825682, 8193544, 37087449, 169114547, 776110247, 3581944258, 16614576945, 77410877233, 362126147797, 1700179143293, 8008689767674, 37838553977426, 179268540549758, 851478474635404, 4053760582437106
Offset: 1

Views

Author

Paul D. Hanna, Aug 27 2022

Keywords

Comments

Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f. A(x) = x + x^3 + 2*x^5 + 6*x^7 + 21*x^9 + 78*x^11 + 303*x^13 + 1223*x^15 + 5085*x^17 + 21623*x^19 + 93585*x^21 + ...
where A(x)^2 = A( x^2 + 2*x^2*A(x)^2 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^4 + 5*x^6 + 16*x^8 + 58*x^10 + 222*x^12 + 882*x^14 + 3616*x^16 + 15205*x^18 + 65220*x^20 + ...
x^2 + 2*x^2*A(x)^2 = x^2 + 2*x^4 + 4*x^6 + 10*x^8 + 32*x^10 + 116*x^12 + 444*x^14 + 1764*x^16 + 7232*x^18 + 30410*x^20 + ...
Let G(x) = Series_Reversion( A(x) ) then
G(x) = x - x^3 + x^5 - 2*x^7 + 4*x^9 - 7*x^11 + 12*x^13 - 23*x^15 + 45*x^17 - 84*x^19 + 157*x^21 - 302*x^23 + 584*x^25 - 1121*x^27 + ...
where G(x)^2 = G(x^2)/(1 + 2*x^2) and G(A(x)) = x.

Crossrefs

Cf. A000108, A356781, A271931, A271932, A271933.

Programs

  • PARI
    {a(n) = my(A=x+x^3, X=x+x*O(x^(2*n))); for(i=1, 2*n, A = subst(A, x, x^2 + 2*X^2*A^2)^(1/2) ); polcoeff(A, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1) satisfies:
(1) A(x) = sqrt( A( x^2 + 2*x^2*A(x)^2 ) ).
(2) G(x) = sqrt( G(x^2) / (1 + 2*x^2) ), where A(G(x)) = x.
Showing 1-3 of 3 results.

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