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 1-10 of 13 results. Next

A178852 G.f. satisfies: A(x) = x/(x - B(x^2)) where B(x/A(x)) = x and B(x) is the g.f. of A141200.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 10, 21, 37, 79, 144, 311, 580, 1262, 2393, 5236, 10055, 22095, 42857, 94495, 184784, 408557, 804331, 1782470, 3529190, 7836235, 15591086, 34676360, 69284645, 154320310, 309480750, 690193910, 1388679639, 3100467566
Offset: 0

Views

Author

Paul D. Hanna, Aug 11 2010

Keywords

Comments

The g.f. of A141200 satisfies: B(x) = x + B(B(x)^2).

Examples

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 21*x^7 +...
If B(x) = g.f. of A141200, with B(x/A(x)) = x and B(x) = x + B(B(x)^2), then
B(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 72*x^6 + 272*x^7 +... where
x/A(x) = x - (x^2 + x^4 + 2*x^6 + 6*x^8 + 20*x^10 + 72*x^12 + 272*x^14 +...)
A(B(x)) = B(x)/x = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 72*x^5 + 272*x^6 +...

Links

Crossrefs

Cf. A141200.

Programs

  • PARI
    {a(n)=local(A=1+x+x^2*O(x^n)); for(i=0,#binary(n)+1, A=x/(x-subst(serreverse(x/A), x, x^2+x^2*O(x^n)))) ; polcoeff(A, n)}
for(n=0,40,print1(a(n),", "))

Formula

a(n) ~ c * d^n / n^(3/2), where d = 2.20085985704067535258..., c = 4.25914484723... if n is even and c = 4.40480643955... if n is odd. - Vaclav Kotesovec, Dec 02 2014

A275755 G.f. satisfies: A(x) = x + A( A(x)^2 - A(x)^5 ).

Original entry on oeis.org

1, 1, 2, 6, 19, 65, 234, 873, 3346, 13099, 52154, 210541, 859768, 3545263, 14741148, 61736903, 260192880, 1102704585, 4696416190, 20090502706, 86285786519, 371917832707, 1608317086940, 6975728777332, 30338392601498, 132277349730004, 578075052215714, 2531710609461484, 11109852467209553, 48843541287179595, 215108137824940916, 948874606956945665, 4191979050580762418, 18545890698661636784, 82159569800859439840, 364432560308538162214, 1618431087549954575022
Offset: 1

Views

Author

Paul D. Hanna, Aug 20 2016

Keywords

Comments

Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 19*x^5 + 65*x^6 + 234*x^7 + 873*x^8 + 3346*x^9 + 13099*x^10 + 52154*x^11 + 210541*x^12 + 859768*x^13 + 3545263*x^14 +...
such that A(x) = x + A( A(x)^2 - A(x)^5 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 54*x^6 + 192*x^7 + 710*x^8 + 2702*x^9 + 10515*x^10 + 41660*x^11 + 167483*x^12 + 681532*x^13 + 2801816*x^14 +...
A(x)^5 = x^5 + 5*x^6 + 20*x^7 + 80*x^8 + 320*x^9 + 1286*x^10 + 5210*x^11 + 21285*x^12 + 87655*x^13 + 363660*x^14 + 1518952*x^15 +...
A(x^2 - x^5) = x^2 + x^4 - x^5 + 2*x^6 - 2*x^7 + 6*x^8 - 6*x^9 + 20*x^10 - 24*x^11 + 71*x^12 - 95*x^13 + 270*x^14 - 392*x^15 + 1063*x^16 - 1662*x^17 +...
where Series_Reversion(A(x)) = x - A(x^2 - x^5).

Links

Crossrefs

Cf. A275756, A275757, A275758, A141200.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = x + subst(A,x, A^2 - A^5 +x*O(x^n))); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

Formula

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

A275756 G.f. satisfies: A(x) = x + A( A(x)^2 - A(x)^6 ).

Original entry on oeis.org

1, 1, 2, 6, 20, 71, 264, 1018, 4032, 16305, 67042, 279444, 1178088, 5014596, 21521488, 93027025, 404630318, 1769704106, 7778030834, 34335337802, 152168657438, 676796514510, 3019945599904, 13515300673984, 60649985907334, 272847379282493, 1230295797205452, 5559373120441048, 25171114275512520, 114177375142080814, 518806321789317040, 2361183952087172306, 10762422470020855820, 49125407360603361370, 224533932290057629076, 1027553322543206612019, 4708070541211739962738, 21595828228486254332762
Offset: 1

Views

Author

Paul D. Hanna, Aug 20 2016

Keywords

Comments

Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
Compare to: C(x) = x + C( C(x)^3 - C(x)^9 ) holds when C(x) = x + C(x)^3 is a g.f. of the ternary tree numbers (A001764).

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 71*x^6 + 264*x^7 + 1018*x^8 + 4032*x^9 + 16305*x^10 + 67042*x^11 + 279444*x^12 + 1178088*x^13 + 5014596*x^14 +...
such that A(x) = x + A( A(x)^2 - A(x)^6 ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 56*x^6 + 206*x^7 + 786*x^8 + 3088*x^9 + 12408*x^10 + 50754*x^11 + 210639*x^12 + 884784*x^13 + 3754424*x^14 +...
A(x)^6 = x^6 + 6*x^7 + 27*x^8 + 116*x^9 + 495*x^10 + 2112*x^11 + 9035*x^12 + 38820*x^13 + 167628*x^14 + 727480*x^15 + 3172455*x^16 +...
A(x^2 - x^6) = x^2 + x^4 + x^6 + 4*x^8 + 14*x^10 + 48*x^12 + 170*x^14 + 628*x^16 + 2382*x^18 + 9202*x^20 + 36098*x^22 + 143484*x^24 + 576638*x^26 + 2339050*x^28 +...
where Series_Reversion(A(x)) = x - A(x^2 - x^6).

Links

Crossrefs

Cf. A275755, A275757, A275758, A141200.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = x + subst(A,x, A^2 - A^6 +x*O(x^n))); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

Formula

G.f. satisfies:
(1) A(x - A(x^2 - x^6)) = x.
(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^2-x^6)^(n+1) / (n+1)!.
(3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^2-x^6)^(n+1)/x / (n+1)! ).

A141201 G.f. satisfies: A(x)^(1/2) = x + A(x) + A(A(x)) + A(A(A(x))) + A(A(A(A(x)))) +...

Original entry on oeis.org

1, 2, 5, 16, 56, 208, 804, 3202, 13046, 54120, 227812, 970596, 4177436, 18136008, 79326641, 349241700, 1546401412, 6882164584, 30767887372, 138114569908, 622270344740, 2812994557488, 12755010576700, 57996678526684
Offset: 2

Views

Author

Paul D. Hanna, Jun 12 2008

Keywords

Examples

			G.f.: A(x) = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 56*x^6 + 208*x^7 +...
A(x)^(1/2) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 72*x^6 + 272*x^7 +...
Related expansions:
A(A(x)) = x^4 + 4*x^5 + 16*x^6 + 64*x^7 + 260*x^8 + 1072*x^9 +...
A(A(A(x))) = x^8 + 8*x^9 + 48*x^10 + 256*x^11 + 1290*x^12 +...
A(A(A(A(x)))) = x^16 + 16*x^17 + 160*x^18 + 1280*x^19 + 8980*x^20 +...
where A(x)^(1/2) = x + A(x) + A(A(x)) + A(A(A(x))) + ...

Links

Crossrefs

Cf. A141200.

Programs

  • PARI
    {a(n)=local(A=x+x^2);for(i=0,n,A=serreverse(x-subst(A,x,x^2+x^2*O(x^n))));polcoeff(A^2,n)}

Formula

G.f.: A(x) = G(x)^2 where G(x) = g.f. of A141200.
a(n) ~ c / (n^(3/2) * r^n), where r = 0.2064501590536889244980412... (see A141200) and c = 0.051350746503831964896433888... . - Vaclav Kotesovec, Dec 02 2014

A179486 G.f. A(x) satisfies A(x) = x + A(A(x)^3) where A(x) = Sum_{n>=1} a(n)*x^(2*n-1).

Original entry on oeis.org

1, 1, 3, 12, 56, 285, 1533, 8571, 49311, 290019, 1735845, 10538550, 64741482, 401708636, 2513837931, 15847467276, 100547976684, 641571037002, 4114313992851, 26503239829953, 171416342026944, 1112726829455289
Offset: 1

Views

Author

Paul D. Hanna, Aug 12 2010

Keywords

Examples

			G.f.: A(x) = x + x^3 + 3*x^5 + 12*x^7 + 56*x^9 + 285*x^11 +...
A(x)^3 = x^3 + 3*x^5 + 12*x^7 + 55*x^9 + 276*x^11 + 1470*x^13 +...
The series reversion of A(x) equals x - A(x^3), therefore
x = A(x - x^3 - x^9 - 3*x^15 - 12*x^21 - 56*x^27 - 285*x^33 -...).
Let G(x) = A(x)^3 be the g.f. of A179487, then
G(G(x)) = x^9 + 9*x^11 + 63*x^13 + 411*x^15 + 2619*x^17 +...,
G(G(G(x))) = x^27 + 27*x^29 + 432*x^31 + 5364*x^33 +..., and
G(G(G(G(x)))) = x^81 + 81*x^83 + 3483*x^85 +...
where A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) +...

Crossrefs

Cf. A179487, A141200 (variant).

Programs

  • Maxima
    Co(n,k,F):=if k=1 then F(n) else sum(F(i+1)*Co(n-i-1, k-1, F),i,0,n-k);
b(n):=if n=1 then 1 else sum(if 3*k>n then 0 else Co(n,3*k,b)*b(k),k,1, n);
a(n):=b(2*n-1);
makelist(a(n),n,1,7); [Vladimir Kruchinin, Jun 28 2011]
  • Maxima
    T(n,m):=if n=m then 1 else kron_delta(n,m)+sum(binomial(m,j)*sum(if 3*k<=n-j then T(n-j,3*k)*T(k,m-j) else 0,k,m-j,n-j),j,0,m-1);
makelist(T(n,1),n,1,12); [Vladimir Kruchinin, May 02 2012]
  • PARI
    {a(n)=local(A=x+x^3); for(i=0, n, A=serreverse(x-subst(A, x, x^3+x^2*O(x^(2*n))))); polcoeff(A, 2*n-1)}

Formula

G.f. A(x) satisfies: A(x - A(x^3)) = x.
G.f.: A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) +... where G(x) = A(x)^3 = g.f. of A179487.
Given g.f. A(x), define C(x) = A(x^3), then C(x) = x^3 + C(C(x)).
Derivative of g.f. A(x) satisfies: A'(x) = 1/(1 - 3*A(x)^2*A'(A(x)^3)).
Radius of convergence, r, and related values:
r = 0.378590868760339249702289974755919481665219504207137681721365...;
A(r) = 0.5510035125320818261355419567786182869427265480378585343298... where r = A(r) - A(A(r)^3);
A(r)^3 = 0.1672873502451522851544780724841939477291722823741494215...;
A(A(r)^3) = 0.1724126437717425764332519820226988052775070438307208...;
A'(A(r)^3) = 1.0979182660346808662695442970765885990300854399844658... where A'(A(r)^3) = 1/(3*A(r)^2);
Limit a(n+1)/a(n) = 1/r^2 = 6.9768555281242291444841704586123374638...
Let V(x) = x/(x - A(x^3)) then V'(A(r)) = 1/r, V(z) - z*V'(z) = 0 at z=A(r), and V(A(x)) = A(x)/x for all x.

Extensions

Typo in example corrected by Paul D. Hanna, Aug 13 2010

A295760 G.f. A(x) satisfies: A(x - A(x^2)) = x + A(x^2).

Original entry on oeis.org

1, 2, 4, 14, 52, 204, 840, 3586, 15708, 70196, 318792, 1467068, 6826360, 32062184, 151805328, 723785606, 3472055348, 16745865716, 81154862712, 394993248572, 1929956966024, 9463011127592, 46547604953424, 229631732978956, 1135868927282840, 5632406184568280, 27992869133724208, 139417039713595032, 695719759056964304, 3478120152935676720, 17417832347321830432
Offset: 1

Views

Author

Paul D. Hanna, Nov 26 2017

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 4*x^3 + 14*x^4 + 52*x^5 + 204*x^6 + 840*x^7 + 3586*x^8 + 15708*x^9 + 70196*x^10 + 318792*x^11 + 1467068*x^12 + 6826360*x^13 + 32062184*x^14 + 151805328*x^15 + 723785606*x^16 + 3472055348*x^17 + 16745865716*x^18 + 81154862712*x^19 + 394993248572*x^20 +...
such that  A(x - A(x^2))  =  x + A(x^2).
RELATED SERIES.
A(x - A(x^2)) = x + x^2 + 2*x^4 + 4*x^6 + 14*x^8 + 52*x^10 + 204*x^12 + 840*x^14 + 3586*x^16 + 15708*x^18 + 70196*x^20 +...
which equals x + A(x^2).
Series_Reversion( x - A(x^2) ) = x + x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 102*x^6 + 420*x^7 + 1793*x^8 + 7854*x^9 + 35098*x^10 +...
which equals (A(x) + x)/2.
A( (x + A(x))^2/4 ) = x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 102*x^6 + 420*x^7 + 1793*x^8 + 7854*x^9 + 35098*x^10 +...
which equals (A(x) - x)/2.
(x + A(x))^2/4 = x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 70*x^6 + 284*x^7 + 1197*x^8 + 5198*x^9 + 23078*x^10 +...
which equals x*A(x) + A( (x + A(x))^2/4 )^2.
Let B(B(x)) = A(x) then B(x) is an integer series (verified up to 400 terms):
B(x) = x + x^2 + x^3 + 4*x^4 + 10*x^5 + 33*x^6 + 105*x^7 + 360*x^8 + 1244*x^9 + 4350*x^10 + 15488*x^11 + 55514*x^12 + 201220*x^13 + 735409*x^14 + 2707973*x^15 + 10036908*x^16 + 37413444*x^17 + 140192022*x^18 + 527728468*x^19 + 1994613008*x^20 + 7566519020*x^21 + 28803657194*x^22 + 110000675444*x^23 + 421172979138*x^24 + 1616154840122*x^25 + 6220675694876*x^26 + 24028744940126*x^27 + 92796758654138*x^28 + 357109198506472*x^29 + 1389208125591993*x^30 + 5552344056227841*x^31 + 21323110914365336*x^32 + 70454013218649400*x^33 + 298345385254918858*x^34 + 2355991303858543108*x^35 + 6997638978589417444*x^36 - 100308304079135213248*x^37 - 153429582192527911554*x^38 + 14890888637965051478428*x^39 + 29224269440712871606248*x^40 +...
where B(x - A(x^2)) is an odd function that begins
B(x - A(x^2)) = x - x^3 - 7*x^5 - 37*x^7 - 189*x^9 - 583*x^11 + 1255*x^13 + 28711*x^15 + 218937*x^17 + 890823*x^19 + 760249*x^21 - 69490111*x^23 + 649102437*x^25 - 54674874881*x^27 + 3035063408777*x^29 - 209576545439765*x^31 + 16859178743641679*x^33 - 1563274578274583407*x^35 + 165493803731758762623*x^37 - 19828488694059640880745*x^39 +...
also B(-B(-x)) = x.

Links

Crossrefs

Cf. A141200.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = x + 2 * A( (x + A(x))^2/4 ).
(2) A(x) = -x + 2 * Series_Reversion( x - A(x^2) ).
(3) x = A( -x + 2 * Series_Reversion( x + A(x^2) ) ).
(4) x*A(x) = (x + A(x))^2/4 - A( (x + A(x))^2/4 )^2.
(5) A(x - A(x^2)) = x + A(x^2).
(6) A(-A(-x)) = x.

A271844 G.f. A(x) satisfies: A(x) = x + A( A(x)^2 + A(x)^4 ).

Original entry on oeis.org

1, 1, 2, 7, 26, 102, 420, 1793, 7854, 35106, 159492, 734334, 3418892, 16068532, 76135112, 363283763, 1744135306, 8419281306, 40838500796, 198950342814, 972999755364, 4775441138580, 23513016382120, 116111875760294, 574927064750460, 2853800953323468, 14197997592237912, 70786396399962476, 353611516341840008, 1769694222850151128
Offset: 1

Views

Author

Paul D. Hanna, Apr 15 2016

Keywords

Comments

Compare g.f. to: C(x) = x + C( C(x)^2 - C(x)^4 ) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 102*x^6 + 420*x^7 + 1793*x^8 + 7854*x^9 + 35106*x^10 + 159492*x^11 + 734334*x^12 +...
where A(x) = x + A( A(x)^2 + A(x)^4 ).
RELATED SERIES.
A(x)^2 + A(x)^4 = x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 84*x^6 + 340*x^7 + 1434*x^8 + 6226*x^9 + 27632*x^10 + 124820*x^11 + 572000*x^12 +...
A(x^2 + x^4) = x^2 + 2*x^4 + 4*x^6 + 14*x^8 + 60*x^10 + 276*x^12 + 1320*x^14 + 6530*x^16 + 33188*x^18 + 172252*x^20 + 909016*x^22 +...
where the series reversion of A(x) equals x - A(x^2 + x^4).

Links

Crossrefs

Cf. A155134, A141200.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A( x - A(x^2 + x^4) ) = x.
(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^2+x^4)^(n+1) / (n+1)!.
(3) A(x) = x*exp( Sum_{n>=0} d^n/dx^n A(x^2+x^4)^(n+1)/x / (n+1)! ).
a(n) ~ c * d^n / n^(3/2), where d = 5.26908951612012208739853420341892... and c = 0.063610446185354820395355587671... . - Vaclav Kotesovec, Apr 16 2016

A272460 G.f. A(x) satisfies: A( x - A(x^3)/x ) = x.

Original entry on oeis.org

1, 1, 2, 5, 15, 49, 168, 596, 2170, 8063, 30451, 116545, 451038, 1762065, 6939684, 27523374, 109832228, 440668881, 1776599145, 7193526536, 29240389629, 119276102017, 488106369196, 2003299984825, 8244088853598, 34010402405020, 140627814353509, 582704045483909, 2419228983607503, 10062353339406026, 41924039720446064, 174952464642171681, 731184941189099208, 3060168941260579386
Offset: 1

Views

Author

Paul D. Hanna, Apr 29 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 49*x^6 + 168*x^7 + 596*x^8 + 2170*x^9 + 8063*x^10 + 30451*x^11 + 116545*x^12 +...
where A( x - A(x^3)/x ) = x.
RELATED SERIES.
Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then
B(x) = x - x^2 - x^5 - 2*x^8 - 5*x^11 - 15*x^14 - 49*x^17 - 168*x^20 - 596*x^23 - 2170*x^26 - 8063*x^29 - 30451*x^32 - 116545*x^35 +...
such that B(x) = x - A(x^3)/x.

Links

Crossrefs

Cf. A141200, A272461.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = serreverse( x - subst(A,x,x^3 +x^3*O(x^n))/x )); polcoeff(A,n)}
for(n=1,50,print1(a(n),", "))

Formula

a(n) ~ c * d^n / n^(3/2), where d = 4.3788729685558146277374586... and c = 0.0933818743555997288781743... . - Vaclav Kotesovec, May 03 2016

A272461 G.f. A(x) satisfies: A( x - A(x^4)/x^2 ) = x.

Original entry on oeis.org

1, 1, 2, 5, 14, 43, 140, 474, 1650, 5865, 21194, 77623, 287492, 1074915, 4051824, 15381073, 58749102, 225621404, 870686810, 3374625925, 13130575110, 51271434788, 200845390668, 789081913225, 3108496250028, 12275905239752, 48590260462470, 192736593501813, 766007363990640, 3049978926971396, 12164745517874576, 48596364360237882, 194426663474794450, 778968350863994065
Offset: 1

Views

Author

Paul D. Hanna, Apr 29 2016

Keywords

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 43*x^6 + 140*x^7 + 474*x^8 + 1650*x^9 + 5865*x^10 + 21194*x^11 + 77623*x^12 +...
where A( x - A(x^4)/x^2 ) = x.
RELATED SERIES.
Let B(x) be the series reversion of the g.f. A(x) so that A(B(x)) = x, then
B(x) = x - x^2 - x^6 - 2*x^10 - 5*x^14 - 14*x^18 - 43*x^22 - 140*x^26 - 474*x^30 - 1650*x^34 - 5865*x^38 - 21194*x^42 - 77623*x^46 +...
such that B(x) = x - A(x^4)/x^2.

Links

Crossrefs

Cf. A141200, A272460.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1,n, A = serreverse( x - subst(A,x,x^4 +x^3*O(x^n))/x^2 )); polcoeff(A,n)}
for(n=1,50,print1(a(n),", "))

Formula

a(n) ~ c * d^n / n^(3/2), where d = 4.1920029654932692520828... and c = 0.1046247209912855075794... . - Vaclav Kotesovec, May 03 2016

A295761 G.f. A(x) satisfies: A(x - A(x^2)) = x + 2*A(x^2).

Original entry on oeis.org

1, 3, 6, 24, 96, 396, 1728, 7839, 36438, 172680, 831624, 4058202, 20021268, 99697188, 500429016, 2529375300, 12862429920, 65760468840, 337817930184, 1742850773154, 9026374329108, 46912014922392, 244588357460448, 1278937818954306, 6705339839722404, 35241796466506908, 185643541655678184, 979972436105339856, 5183169679909147200, 27464173024052341200
Offset: 1

Views

Author

Paul D. Hanna, Dec 03 2017

Keywords

Examples

			G.f.: A(x) = x + 3*x^2 + 6*x^3 + 24*x^4 + 96*x^5 + 396*x^6 + 1728*x^7 + 7839*x^8 + 36438*x^9 + 172680*x^10 + 831624*x^11 + 4058202*x^12 +...
such that  A(x - A(x^2))  =  x + 2*A(x^2).
RELATED SERIES.
A(x - A(x^2)) = x + 2*x^2 + 6*x^4 + 12*x^6 + 48*x^8 + 192*x^10 + 792*x^12 + 3456*x^14 + 15678*x^16 + 72876*x^18 + 345360*x^20 + 1663248*x^22 + 8116404*x^24 + 40042536*x^26 + 199394376*x^28 + 1000858032*x^30 + 5058750600*x^32 +...
which equals x + 2*A(x^2).
Series_Reversion( x - A(x^2) ) = x + x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 132*x^6 + 576*x^7 + 2613*x^8 + 12146*x^9 + 57560*x^10 + 277208*x^11 + 1352734*x^12 +...
which equals (A(x) + 2*x)/3.
A( (2*x + A(x))^2/9 ) = x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 132*x^6 + 576*x^7 + 2613*x^8 + 12146*x^9 + 57560*x^10 + 277208*x^11 + 1352734*x^12 +...
which equals (A(x) - x)/3.

Links

Crossrefs

Cf. A141200, A295760, A295762.

Programs

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

Formula

G.f. A(x) satisfies:
(1) A(x) = x + 3 * A( (2*x + A(x))^2/9 ).
(2) A(x) = -2*x + 3 * Series_Reversion( x - A(x^2) ).
(3) x = A( -x/2 + 3/2 * Series_Reversion( x + 2*A(x^2) ) ).
(4) A(x - A(x^2)) = x + 2*A(x^2).
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