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 1-7 of 7 results.

A214372 G.f. satisfies A(x) = x + A(x)^2*(1 + A(x))^2.

Original entry on oeis.org

1, 1, 4, 16, 74, 364, 1876, 9993, 54582, 304040, 1720576, 9864114, 57169168, 334404892, 1971641096, 11705119157, 69911438676, 419798442416, 2532791105844, 15346498242460, 93344296450020, 569741502841020, 3488539758461880, 21422341792366320
Offset: 1

Views

Author

Paul D. Hanna, Jul 14 2012

Keywords

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 364*x^6 + 1876*x^7 +...
Related expansions:
A(x) = x + x^2*(1+x)^2 + d/dx x^4*(1+x)^4/2! + d^2/dx^2 x^6*(1+x)^6/3! + d^3/dx^3 x^8*(1+x)^8/4! +...
log(A(x)/x) = x*(1+x)^2 + d/dx x^3*(1+x)^4/2! + d^2/dx^2 x^5*(1+x)^6/3! + d^3/dx^3 x^7*(1+x)^8/4! +...
A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 40*x^5 + 196*x^6 + 1004*x^7 + 5328*x^8 +...
(1+A(x))^2 = 1 + 2*x + 3*x^2 + 10*x^3 + 41*x^4 + 188*x^5 + 924*x^6 + 4756*x^7 + 25314*x^8 +...
Series reversion shows a relation to the Catalan numbers (A000108):
Series_Reversion( sqrt(A(x) - x) ) =  (sqrt(1+4*x) - 1)/2 - x^2, which begins
x - 2*x^2 + 2*x^3 - 5*x^4 + 14*x^5 - 42*x^6 + 132*x^7 - 429*x^8 +...
where
sqrt(A(x) - x) = x + 2*x^2 + 6*x^3 + 25*x^4 + 114*x^5 + 560*x^6 + 2880*x^7 +...+ A229042(n)*x^n +...

Links

Crossrefs

Cf. A229042, A361305, A361306.

Programs

  • Maple
    a:= n-> coeff(series(RootOf(A=x+A^2*(1+A)^2, A), x, n+1), x, n):
seq(a(n), n=1..30);  # Alois P. Heinz, Feb 13 2017
  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x - x^2*(1+x)^2,{x,0,20}],x],x]] (* Vaclav Kotesovec, Sep 17 2013 *)
  • PARI
    {a(n)=polcoeff(serreverse(x-x^2*(1+x)^2+x*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(2*m)/m!)); polcoeff(A, n)}
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*(1+x+x*O(x^n))^(2*m)/m!))); polcoeff(A, n)}
  • PARI
    {a(n)=polcoeff(x + serreverse( (sqrt(1+4*x +x*O(x^n)) - 1)/2 - x^2 )^2,n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = Series_Reversion( x - x^2*(1+x)^2 ).
(2) A(x) = x + Series_Reversion( (sqrt(1+4*x) - 1)/2 - x^2 )^2. - Paul D. Hanna, Oct 24 2013
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)*(1+x)^(2*n)/n!.
(4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1+x)^(2*n)/n! ).
Recurrence: 31*(n-2)*(n-1)*n*(104*n - 293)*a(n) = 8*(n-2)*(n-1)*(1768*n^2 - 7633*n + 7425)*a(n-1) + 20*(n-2)*(2080*n^3 - 14180*n^2 + 31612*n - 22995)*a(n-2) + 8*(2*n-5)*(4*n-13)*(4*n-11)*(104*n - 189)*a(n-3). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ 1/312*sqrt(78)*sqrt((26533 + 50583*sqrt(78))^(2/3) - 5837 + 13*(26533 + 50583*sqrt(78))^(1/3))/((26533 + 50583*sqrt(78))^(1/6)) * (4/93*((209773 + 4836*sqrt(78))^(2/3) + 3481 + 34*(209773 + 4836* sqrt(78))^(1/3))/(209773 + 4836*sqrt(78))^(1/3))^n / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 17 2013
a(n+1) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(2*k,n-k)/(n+k+1). - Seiichi Manyama, Aug 24 2023

A361306 Expansion of A(x) satisfying A(x) = x + A(x)^2*(1 + A(x))^4.

Original entry on oeis.org

1, 1, 6, 31, 186, 1191, 7972, 55164, 391322, 2830751, 20801826, 154853413, 1165316224, 8850372878, 67750780816, 522218420336, 4049564739054, 31570368061361, 247293510244174, 1945331619223591, 15361731119713506, 121729460653957980, 967664450692965300
Offset: 1

Views

Author

Paul D. Hanna, Mar 08 2023

Keywords

Examples

			G.f.: A(x) = x + x^2 + 6*x^3 + 31*x^4 + 186*x^5 + 1191*x^6 + 7972*x^7 + 55164*x^8 + 391322*x^9 + ...
such that sqrt(A(x) - x) = A(x)*(1 + A(x))^2.
A(x)*(1 + A(x))^2 = x + 3*x^2 + 11*x^3 + 60*x^4 + 355*x^5 + 2261*x^6 + 15094*x^7 + 104208*x^8 + ...
A(x)*(1 + A(x))^2 = Series_Reversion( -x^2 + Sum_{n>=1} (-1)^(n-1) * binomial(3*n-2,n-1)*x^n/n ).

Links

Crossrefs

Cf. A361304, A361305, A214372, A006013.

Programs

  • PARI
    {a(n)=polcoeff(serreverse(x - x^2*(1+x)^4 +x*O(x^n)), n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(4*m)/m!)); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*(1+x+x*O(x^n))^(4*m)/m!))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following.
(1) A(x) = Series_Reversion( x - x^2*(1+x)^4 ).
(2) A(x) = x + A(x)^2 * (1 + A(x))^4.
(3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n) * (1+x)^(4*n) / n!.
(4) A(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(4*n) / n! ).
(5) A(x) = x + Series_Reversion( Series_Reversion( x*(1+x)^2 ) - x^2 )^2.
(6) A(x) = x + Series_Reversion( -x^2 + Sum_{n>=1} (-1)^(n-1) * binomial(3*n-2,n-1) * x^n/n )^2.
From Vaclav Kotesovec, Mar 09 2023: (Start)
Recurrence: 3381*(n-4)*(n-3)*(n-2)*(n-1)*n*(4485934293448*n^5 - 88905588075732*n^4 + 698950092208066*n^3 - 2724285958475163*n^2 + 5263801532363671*n - 4032831805999290)*a(n) = 2*(n-4)*(n-3)*(n-2)*(n-1)*(33204885640102096*n^6 - 707886491396721408*n^5 + 6160367858867908768*n^4 - 27918165429184721124*n^3 + 69150795811214975011*n^2 - 88077097294043237943*n + 44480953779348451050)*a(n-1) + 8*(n-4)*(n-3)*(n-2)*(52772531028122272*n^7 - 1256975462235400336*n^6 + 12611049851568548176*n^5 - 69004162305753446968*n^4 + 222104765912229832762*n^3 - 419924105934755620321*n^2 + 431120275047208552290*n - 185089750933520270250)*a(n-2) + 48*(n-4)*(n-3)*(17647665510424432*n^8 - 482112074818112928*n^7 + 5693971809001104840*n^6 - 37956706633792772384*n^5 + 156126872715173363823*n^4 - 405548028261835673882*n^3 + 649232078072133939050*n^2 - 585187986606994739801*n + 227161430445970883100)*a(n-3) + 32*(n-4)*(21945190563547616*n^9 - 698268423629052336*n^8 + 9788485232517982416*n^7 - 79313303231764021176*n^6 + 409187506797434806734*n^5 - 1393249646753024170299*n^4 + 3129189249705937191544*n^3 - 4467594298222926610959*n^2 + 3676695031470911619960*n - 1327813620065788842000)*a(n-4) + 72*(2*n - 7)*(3*n - 14)*(3*n - 13)*(6*n - 31)*(6*n - 29)*(4485934293448*n^5 - 66475916608492*n^4 + 388187082839618*n^3 - 1116009867370877*n^2 + 1578887211201855*n - 878785793685000)*a(n-5).
a(n) ~ 1/(2 * (1 + s) * sqrt(Pi*(1 + 10*s + 15*s^2)) * n^(3/2) * r^(n - 1/2)), where r = 0.1176087332021218420455915375218722861407778043565... and s = 0.1894485384658193296593809633217117092941452563863... are real roots of the system of equations r + s^2 * (1+s)^4 = s, 2*s*(1+s)^3 * (1+3*s) = 1. (End)
a(n+1) = Sum_{k=0..n} binomial(n+k+1,k) * binomial(4*k,n-k)/(n+k+1). - Seiichi Manyama, Aug 24 2023

A365157 G.f. satisfies A(x) = ( 1 + x*A(x)^2*(1 + x*A(x))^3 )^2.

Original entry on oeis.org

1, 2, 15, 124, 1167, 11772, 124561, 1363964, 15326826, 175739698, 2047974619, 24185317182, 288801732423, 3481242975808, 42303574158234, 517683469595912, 6374096109874427, 78909384182870688, 981600144994348111, 12263583888826309544, 153812133876403777005
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2023

Keywords

Crossrefs

Cf. A361305, A365158.

Programs

  • PARI
    a(n, s=3, t=2) = sum(k=0, n, binomial(t*(n+k+1), k)*binomial(s*k, n-k)/(n+k+1));

Formula

If g.f. satisfies A(x) = ( 1 + x*A(x)^2*(1 + x*A(x))^s )^t, then a(n) = Sum_{k=0..n} binomial(t*(n+k+1),k) * binomial(s*k,n-k)/(n+k+1).

A365158 G.f. satisfies A(x) = ( 1 + x*A(x)^2*(1 + x*A(x))^3 )^3.

Original entry on oeis.org

1, 3, 30, 352, 4680, 66852, 1002420, 15562917, 248028012, 4034367018, 66704722941, 1117794312987, 18942067925094, 324048616144950, 5588890522700901, 97074537335184054, 1696556614819124517, 29812650855663860436, 526429300730659123740
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2023

Keywords

Crossrefs

Cf. A361305, A365157.

Programs

  • PARI
    a(n, s=3, t=3) = sum(k=0, n, binomial(t*(n+k+1), k)*binomial(s*k, n-k)/(n+k+1));

Formula

If g.f. satisfies A(x) = ( 1 + x*A(x)^2*(1 + x*A(x))^s )^t, then a(n) = Sum_{k=0..n} binomial(t*(n+k+1),k) * binomial(s*k,n-k)/(n+k+1).

A361303 Expansion of g.f. A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(3*n) / n!.

Original entry on oeis.org

1, 2, 15, 92, 615, 4200, 29190, 205416, 1458909, 10436030, 75079719, 542669244, 3937604853, 28664996080, 209261546580, 1531373181120, 11230365782130, 82512324300222, 607246350958449, 4475646134515360, 33031356134381220, 244073892799489500, 1805479496422561740
Offset: 0

Views

Author

Paul D. Hanna, Mar 08 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 15*x^2 + 92*x^3 + 615*x^4 + 4200*x^5 + 29190*x^6 + 205416*x^7 + 1458909*x^8 + 10436030*x^9 + ...

Crossrefs

Cf. A361305, A215128, A361046.

Programs

  • PARI
    {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=1); A = sum(m=0, n, Dx(m, x^(2*m)*(1+x +O(x^(n+1)))^(3*m)/m!)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* Using series reversion (faster) */
{a(n) = my(A=1); A = deriv( serreverse(x - x^2*(1+x +O(x^(n+3)))^3 )); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) A(x) = Sum_{n>=0} d^n/dx^n x^(2*n) * (1 + x)^(3*n) / n!.
(2) A(x) = d/dx Series_Reversion(x - x^2*(1 + x)^3).
(3) B(x - x^2*A(x)^3) = x where B(x) = x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1) * (1+x)^(3*n) / n! ) is the g.f. of A361305.
(4) a(n) = (n+1) * A361305(n+1) for n >= 0.

A367260 G.f. satisfies A(x) = 1 + x*A(x)^3 * (1 + x*A(x))^3.

Original entry on oeis.org

1, 1, 6, 36, 251, 1881, 14817, 120950, 1014042, 8680377, 75552553, 666614637, 5948817600, 53599239101, 486926148000, 4455202562652, 41018936164660, 379747493741643, 3532914858433284, 33012260400580342, 309692626084981245, 2915659701275923491
Offset: 0

Views

Author

Seiichi Manyama, Nov 11 2023

Keywords

Crossrefs

Cf. A019497, A361305, A364742.
Cf. A367233.

Programs

  • PARI
    a(n, s=3, t=3, u=1) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).

A367285 G.f. satisfies A(x) = 1 + x*A(x)^2 * (1 + x*A(x)^2)^3.

Original entry on oeis.org

1, 1, 5, 26, 159, 1042, 7185, 51340, 376806, 2823734, 21516113, 166196703, 1298413089, 10241803340, 81454834164, 652465062453, 5259084437170, 42624217133130, 347160390473763, 2839928983316595, 23323730673818467, 192237734035157372, 1589602164422747636
Offset: 0

Views

Author

Seiichi Manyama, Nov 12 2023

Keywords

Crossrefs

Cf. A360076, A361305.
Cf. A137953, A367261.
Cf. A367240.

Programs

  • PARI
    a(n, s=3, t=2, u=2) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));

Formula

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).
Showing 1-7 of 7 results.

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