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.

A375453 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^3 ).

Original entry on oeis.org

1, 3, 9, 31, 117, 459, 1835, 7449, 30711, 128601, 546537, 2354139, 10260492, 45173868, 200578692, 896865572, 4033380894, 18224524458, 82664886074, 376161628302, 1716301466139, 7848924260901, 35966629306221, 165109474283847, 759210907786198, 3496438156668822, 16126158739138860
Offset: 1

Views

Author

Paul D. Hanna, Aug 16 2024

Keywords

Comments

Compare to: F(x)^2 = F( x^2/(1-2*x) ), where F(x) = x*M(x) and M(x) = 1 + x*M(x) + x^2*M(x)^2 is the Motzkin function (A001006).
Compare to: G(x)^2 = G( x^2/(1-2*x)^2 ), where G(x) = x*C(x)^2 and C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

Examples

			G.f.: A(x) = x + 3*x^2 + 9*x^3 + 31*x^4 + 117*x^5 + 459*x^6 + 1835*x^7 + 7449*x^8 + 30711*x^9 + 128601*x^10 + ...
where A(x)^2 = A( x^2/(1-2*x)^3 ).
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 27*x^4 + 116*x^5 + 501*x^6 + 2178*x^7 + 9491*x^8 + 41424*x^9 + 181293*x^10 + ...
(A(x)/x)^(1/3) = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 77*x^5 + 290*x^6 + 1122*x^7 + 4462*x^8 + 18210*x^9 + ... + A375443(n)*x^n + ...
x/Series_Reversion( A( x^2/(1-2*x) )^(1/2) ) = 1 + x + 2*x^2 - 2*x^4 + 6*x^6 - 20*x^8 + 70*x^10 - 263*x^12 + 1044*x^14 - 4263*x^16 + 17762*x^18 + ...
x/Series_Reversion( A( x^3/(1-2*x)^3 )^(1/3) ) = 1 + 2*x + x^3 - x^6 + 3*x^9 - 10*x^12 + 34*x^15 - 124*x^18 + 482*x^21 - 1931*x^24 + 7893*x^27 + ...
SPECIFIC VALUES.
A(t) = 3/4 at t = 0.201772636312778304679687617697508690090653188...
A(t) = 3/5 at t = 0.194614960496736155296642077884228463225576089...
A(t) = 1/2 at t = 0.186135869221980538627401571340819246192140850...
A(t) = 2/5 at t = 0.173143830263370608074654087902797631449309857...
A(t) = 1/4 at t = 0.140069990039210460387276300843591158073987855...
A(1/5) = 0.700768312277362449514797370811301885385349818...
where A(1/5)^2 = A(5/27).
A(1/6) = 0.362320684925221039201199651574198595785551012...
where A(1/6)^2 = A(6/64).
A(1/7) = 0.259569089568076471080673806323871020166140312...
where A(1/7)^2 = A(7/125).
A(1/10) = 0.14404022241542053703979110789205898915122135...
where A(1/10)^2 = A(10/512).

Links

Crossrefs

Cf. A001006, A000108, A375454, A375455, A375456.
Cf. A375443.

Programs

  • PARI
    {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^3 ) - Ax^2, #A) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x^2/(1-2*x)^3 ).
(2) A(x)^4 = A( x^4*(1-2*x)^3 / ((1-4*x)^3*(1 - 2*x + 2*x^2)^3) ).
(3) A(x^2 + 2*x^3) = A( x/(1+2*x) )^2.
The radius of convergence r satisfies r = (1 - 2*r)^3, where A(r) = 1 and r = (1/12)*(6 + (6*sqrt(87) - 54)^(1/3) - (6*sqrt(87) + 54)^(1/3)) = 0.20512274384927080786...

A375455 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^5 ).

Original entry on oeis.org

1, 5, 20, 90, 470, 2566, 13885, 74435, 400530, 2183930, 12112167, 68351005, 392055575, 2281947435, 13450584580, 80110426698, 481032299830, 2905955107950, 17629836770715, 107254895106265, 653597751574541, 3986386422481665, 24321398369358070, 148386468804372420, 905156432977350225
Offset: 1

Views

Author

Paul D. Hanna, Aug 17 2024

Keywords

Comments

Compare to: F(x)^2 = F( x^2/(1-2*x) ), where F(x) = x*M(x) and M(x) = 1 + x*M(x) + x^2*M(x)^2 is the Motzkin function (A001006).
Compare to: G(x)^2 = G( x^2/(1-2*x)^2 ), where G(x) = x*C(x)^2 and C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

Examples

			G.f.: A(x) = x + 5*x^2 + 20*x^3 + 90*x^4 + 470*x^5 + 2566*x^6 + 13885*x^7 + 74435*x^8 + 400530*x^9 + 2183930*x^10 + ...
where A(x)^2 = A( x^2/(1-2*x)^5 ).
RELATED SERIES.
A(x)^2 = x^2 + 10*x^3 + 65*x^4 + 380*x^5 + 2240*x^6 + 13432*x^7 + 80330*x^8 + 474960*x^9 + 2783590*x^10 + ...
(A(x)/x)^(1/5) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 205*x^5 + 989*x^6 + 4785*x^7 + 23881*x^8 + 124245*x^9 + 673020*x^10 + ...
x/Series_Reversion( A( x^5/(1-2*x)^5 )^(1/5) ) = 1 + 2*x + x^5 - 3*x^10 + 18*x^15 - 124*x^20 + 925*x^25 - 7372*x^30 + 61466*x^35 - 528678*x^40 + 4656736*x^45 + ...
SPECIFIC VALUES.
A(t) = 3/4 at t = 0.1535987460670222421700476984635848015956844093413...
A(t) = 3/5 at t = 0.1494534252284609931621062683479802340037678508370...
A(t) = 1/2 at t = 0.1443598468225794843508026942502138500132562159005...
A(t) = 2/5 at t = 0.1362812665991487577089709044456104123756230678872...
A(t) = 1/4 at t = 0.1144692674833411472616636812900607840273720167873...
A(1/7) = 0.477612316813393143429515106540189409592882329142...
where A(1/7)^2 = A(343/3125).
A(1/8) = 0.309560069127977498956512592550239740786137843207...
where A(1/8)^2 = A(16/243).
A(1/9) = 0.234151149075763124751821214511435118422621268792...
where A(1/9)^2 = A(729/16807).
A(1/10) = 0.189302960006249918030251616127177047165765112599...
where A(1/10)^2 = A(125/4096).

Links

Crossrefs

Cf. A001006, A000108, A375453, A375454, A375456.

Programs

  • PARI
    {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^5 ) - Ax^2, #A) ); H=Ax; A[n+1]}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x^2/(1-2*x)^5 ).
(2) A(x)^4 = A( x^4*(1-2*x)^15 / ((1-2*x)^5 - 2*x^2)^5 ).
(3) A(x^2*(1 + 2*x)^3) = A( x/(1+2*x) )^2.
The radius of convergence r satisfies r = (1 - 2*r)^5, where A(r) = 1 and r = 0.155430268881057887439965...

A375444 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^4 )/(1-2*x).

Original entry on oeis.org

1, 1, 2, 7, 30, 130, 561, 2460, 11115, 51948, 250551, 1240828, 6274580, 32231322, 167460901, 876998437, 4617448333, 24395086617, 129162020323, 684753458054, 3633159683023, 19287528099428, 102441443882448, 544372928359375, 2894576197980724, 15402989792369740, 82040643327234351
Offset: 0

Views

Author

Paul D. Hanna, Aug 19 2024

Keywords

Comments

Compare to M(x)^2 = M( x^2/(1-2*x) )/(1-2*x), where M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of the Motzkin numbers (A001006).
Compare to C(x)^2 = C( x^2/(1-2*x)^2 )/(1-2*x), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 30*x^4 + 130*x^5 + 561*x^6 + 2460*x^7 + 11115*x^8 + 51948*x^9 + 250551*x^10 + ...
where A(x)^2 = A( x^2/(1-2*x)^4 )/(1-2*x).
RELATED SERIES.
A(x)^2 = 1 + 2*x + 5*x^2 + 18*x^3 + 78*x^4 + 348*x^5 + 1551*x^6 + 6982*x^7 + 32114*x^8 + 151620*x^9 + 734458*x^10 + ...
A(x)^4 = 1 + 4*x + 14*x^2 + 56*x^3 + 253*x^4 + 1188*x^5 + 5598*x^6 + 26456*x^7 + 126278*x^8 + ... + A375454(n+1)*x^n + ...
SPECIFIC VALUES.
Given the radius of convergence r = 0.17610056436947880725475...,
A(r) = 1.5436890126920763615708559718017479865252032976509...
  where r = (1-2*r)^4 and A(r) = 1/(1-2*r).
A(1/6) = 1.35888986768048814311476385141914227984504826245...
  where A(1/6)^2 = (3/2)*A(9/64).
A(1/7) = 1.23858760007712401376241920277473621006326963714...
  where A(1/7)^2 = (7/5)*A(49/625).
A(1/8) = 1.18621527667665867031082807873688257681814274612...
  where A(1/8)^2 = (4/3)*A(4/81).
A(1/9) = 1.15430486498931766438966249826580193821574473318...
  where A(1/9)^2 = (9/7)*A(81/2401).
A(1/10) = 1.1323205915354275720071052412999606676975412945...
  where A(1/10)^2 = (5/4)*A(25/1024).

Links

Crossrefs

Cf. A001006, A000108, A375454, A375443, A375445.

Programs

  • Mathematica
    terms = 27; A[] = 1; Do[A[x]=Sqrt[A[x^2/(1-2*x)^4 ]/(1-2*x)] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Aug 21 2025 *)
  • PARI
    {a(n) = my(A=[1], Ax=x); for(i=1, n, A = concat(A, 0); Ax=Ser(A);
A[#A] = (1/2)*polcoeff( subst(Ax, x, x^2/(1-2*x)^4 )/(1-2*x) - Ax^2, #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x^2/(1-2*x)^4 )/(1-2*x).
(2) A(x)^4 = A( x^4*y^4 )*y where y = (1-2*x)^2/((1-2*x)^4 - 2*x^2).
(3) A(x^2 + 4*x^3 + 4*x^4) = A( x/(1+2*x) )^2 / (1+2*x).
The radius of convergence r satisfies r = (1 - 2*r)^4, where A(r) = 1/(1-2*r) and r = 0.17610056436947880725475085178711534652...
Showing 1-3 of 3 results.

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