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.

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...

A375454 Expansion of g.f. A(x) satisfying A(x)^2 = A( x^2/(1-2*x)^4 ).

Original entry on oeis.org

1, 4, 14, 56, 253, 1188, 5598, 26456, 126278, 611640, 3010948, 15058064, 76399263, 392524428, 2038142346, 10674131464, 56281299098, 298286598920, 1586959692508, 8466559886448, 45260129274602, 242296862848648, 1298487003814300, 6964374684442416, 37378818578434617, 200745991803248388
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 + 4*x^2 + 14*x^3 + 56*x^4 + 253*x^5 + 1188*x^6 + 5598*x^7 + 26456*x^8 + 126278*x^9 + 611640*x^10 + ...
where A(x)^2 = A( x^2/(1-2*x)^4 ).
RELATED SERIES.
A(x)^2 = x^2 + 8*x^3 + 44*x^4 + 224*x^5 + 1150*x^6 + 5968*x^7 + 30920*x^8 + 159296*x^9 + 818013*x^10 + ...
(A(x)/x)^(1/2) = 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)/x)^(1/4) = 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 + ...
x/Series_Reversion( A( x^4/(1-2*x)^4 )^(1/4) ) = 1 + 2*x + x^4 - 2*x^8 + 9*x^12 - 46*x^16 + 251*x^20 - 1467*x^24 + 9001*x^28 - 56961*x^32 + 369035*x^36 + ...
SPECIFIC VALUES.
A(t) = 3/4 at t = 0.1736940609090204697398931237543698538457793088071...
A(t) = 3/5 at t = 0.1683900940132911881249251740473132322447213579710...
A(t) = 1/2 at t = 0.1619792168710406277922262531667140037108384145069...
A(t) = 2/5 at t = 0.1519644942152899698264943158671103683281536948439...
A(t) = 1/4 at t = 0.1256102247935771858127425480259396391143977190820...
A(1/6) = 0.56831064552606196969057398988138945640151282529741...
where A(1/6)^2 = A(9/64).
A(1/7) = 0.33620864108171743638518920200481354359529601205566...
where A(1/7)^2 = A(49/625).
A(1/8) = 0.24749365203461325611367946855098276802746512221191...
where A(1/8)^2 = A(4/81).
A(1/9) = 0.19726025709005021216217281111162473370492543981591...
where A(1/9)^2 = A(81/2401).
A(1/10) = 0.1643908422523431149995416239752018754267879073230...
where A(1/10)^2 = A(25/1024).

Links

Crossrefs

Cf. A001006, A000108, A375453, A375455, 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)^4 ) - 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)^4 ).
(2) A(x)^4 = A( x^4*(1-2*x)^8/((1-2*x)^4 - 2*x^2)^4 ).
(3) A(x^2 + 4*x^3 + 4*x^4) = A( x/(1+2*x) )^2.
The radius of convergence r satisfies r = (1 - 2*r)^4, where A(r) = 1 and r = 0.17610056436947880725475...

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...

A259606 G.f. satisfies: A(x) = Series_Reversion( x - A'(x)*A(x)^2 ).

Original entry on oeis.org

1, 1, 6, 60, 790, 12488, 226176, 4567245, 101057170, 2421311002, 62292579316, 1709994461396, 49844902545256, 1536870296603860, 49965056185462360, 1708221871912841430, 61272046476315041664, 2301058164207089144028, 90309756129843950212480, 3697832634432220792202296
Offset: 1

Views

Author

Paul D. Hanna, Jul 02 2015

Keywords

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 60*x^4 + 790*x^5 + 12488*x^6 + 226176*x^7 + 4567245*x^8 + 101057170*x^9 + 2421311002*x^10 + ...
where
A(x - A'(x)*A(x)^2) = x.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 13*x^4 + 132*x^5 + 1736*x^6 + 27276*x^7 + 490408*x^8 + 9831498*x^9 + 216085602*x^10 + ...
A'(x)*A(x)^2 = x^2 + 4*x^3 + 35*x^4 + 434*x^5 + 6664*x^6 + 119072*x^7 + 2392326*x^8 + 52919810*x^9 + 1271042344*x^10 + ...

Links

Crossrefs

Cf. A360578, A375456.

Programs

  • Mathematica
    a[n_] := Module[{A = x}, Do[A = InverseSeries[x - D[A, x] A^2 + x O[x]^n, x], {n}]; SeriesCoefficient[A, {x, 0, n}]];
Array[a, 25] (* Jean-François Alcover, Sep 28 2020, after PARI *)
  • PARI
    {a(n) = local(A=x); for(i=1,n,A=serreverse(x - A^2*A' +x*O(x^n))); polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x, B=x^2); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, (A')^m*A^(2*m)/m!)) +O(x^(n+1))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x, B=x^2); for(i=1, n, B=intformal(2*A); A = x*exp(sum(m=1, n, Dx(m-1, (A')^m*A^(2*m)/(m!*x))) +O(x^(n+1)))); polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. satisfies:
(1) A(x) = x + A'(A(x)) * A(A(x))^2.
(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A'(x)^n * A(x)^(2*n) / n!.
(3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A'(x)^n * A(x)^(2*n) / (n!*x) ).
a(n) ~ c * n! * n^(alfa) / LambertW(1)^n, where alfa = 2.750027682144251700567... and c = 0.005275216890926771261... - Vaclav Kotesovec, Aug 25 2017
alfa = 5*LambertW(1) - 2 + 3/(1 + LambertW(1)). - Vaclav Kotesovec, Mar 13 2023
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