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 1-5 of 5 results.

A371713 Expansion of g.f. A(x) satisfying A(x)^2 = A( A(x)*(x + A(x)^2) ).

Original entry on oeis.org

1, 1, 3, 10, 39, 161, 698, 3126, 14361, 67287, 320319, 1544894, 7532756, 37070678, 183892128, 918539002, 4615979653, 23321497085, 118391352149, 603585987830, 3089089467145, 15864868600157, 81737410659710, 422342729686590, 2188088882282654, 11363944086758244, 59152933495794684
Offset: 1

Views

Author

Paul D. Hanna, Apr 05 2024

Keywords

Examples

			G.f.: A(x) = x + x^2 + 3*x^3 + 10*x^4 + 39*x^5 + 161*x^6 + 698*x^7 + 3126*x^8 + 14361*x^9 + 67287*x^10 + 320319*x^11 + 1544894*x^12 + ...
where A(x)^2 = A( A(x)*(x + A(x)^2) ).
RELATED SERIES.
(1) A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 107*x^6 + 460*x^7 + 2052*x^8 + 9394*x^9 + 43903*x^10 + 208570*x^11 + 1004263*x^12 + ...
(2) A(x)*(x + A(x)^2) = x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 88*x^6 + 374*x^7 + 1652*x^8 + 7512*x^9 + 34920*x^10 + 165198*x^11 + 792700*x^12 + ...
(3) Let R(x) be the series reversion of A(x), R(A(x)) = x, then
R(x) = x - x^2 - x^3 - x^5 - x^9 - x^17 - x^33 - x^65 + ... + -x^(2^n+1) + ...
and R(x) = R(x^2)/x - x^2.
SPECIFIC VALUES.
A(1/6) = 0.2367013365733826841498068726305704943941...
A(1/7) = 0.1823951399847440022737563157206822905959...
A(1/8) = 0.1515149787834965771672802816841610180120...
A(1/9) = 0.1303567976332909027691102900878848253626...
A(1/6)^2 = A(t) at t = A(1/6)*(1/6 + A(1/6)^2) = 0.05271201227864865...
A(1/7)^2 = A(t) at t = A(1/7)*(1/7 + A(1/7)^2) = 0.03212436773155026...

Links

Crossrefs

Cf. A356782, A370439, A372578.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = x + Sum[A[x]^(2^k + 1), {k, 0, Log[nmax]/Log[2]}] + O[x]^(nmax + 1) // Normal, nmax + 1]; Rest[CoefficientList[A[x], x]] (* Vaclav Kotesovec, Apr 05 2024 *)
  • PARI
    /* G.f. Series_Reversion(x - x*Sum_{n>=0} x^(2^n)) */
{a(n) = my(A = serreverse(x - x*sum(k=0,#binary(n), x^(2^k)) +x*O(x^n)));  polcoeff(A,n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[1],F); for(i=1,n, A=concat(A,0); F=x*Ser(A);
A[#A] = -polcoeff( F^2 - subst(F,x, F*(x + F^2)), #A+1) ); A[n]}
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*A(x) + A(x)^3 ).
(2) A(x)^4 = A( x*A(x)^3 + A(x)^5 + A(x)^6 ).
(3) A(x)^8 = A( x*A(x)^7 + A(x)^9 + A(x)^10 + A(x)^12 ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1) + Sum_{k=0..n-1} A(x)^(2^n+2^k) ) for n > 0.
(5) A(x) = x + Sum_{n>=0} A(x)^(2^n+1).
(6) A(x) = Series_Reversion(x - x*Sum_{n>=0} x^(2^n) ).
a(n) ~ c * d^n / n^(3/2), where d = 5.51142100999137014688261137378225123402050823381269982231021216596989145... and c = 0.07924552169373639393012621342284829291839319195254975892205166214809... - Vaclav Kotesovec, Apr 05 2024
The radius of convergence r = 0.1814414101530... = 1/d (d is given above) and A(r) satisfy: 1 = Sum_{n>=0} (2^n+1) * A(r)^(2^n) and r = A(r) - Sum_{n>=0} A(r)^(2^n+1), where A(r) = 0.319865507392391473797021103685180915354570766210154873070... - Paul D. Hanna, Apr 05 2024
c = sqrt(r) / sqrt(2*Pi * Sum_{k>=0} 2^k * (1 + 2^k) * A(r)^(2^k - 1)). - Vaclav Kotesovec, Apr 05 2024

A370439 Expansion of g.f. A(x) satisfying A(x) = A( x*A(x)^2 + 3*x*A(x)^3 )^(1/3).

Original entry on oeis.org

1, 3, 9, 30, 126, 648, 3591, 19953, 110079, 610500, 3440493, 19742616, 114918138, 675417474, 3996992547, 23791052862, 142393544757, 856746349992, 5179722791274, 31449875426622, 191678795532801, 1172198278949454, 7190652243631437, 44235165115911312, 272837082264574914
Offset: 1

Views

Author

Paul D. Hanna, Mar 27 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 + 3*x^2 + 9*x^3 + 30*x^4 + 126*x^5 + 648*x^6 + 3591*x^7 + 19953*x^8 + 110079*x^9 + 610500*x^10 + 3440493*x^11 + 19742616*x^12 + ...
where A(x)^3 = A( x*A(x)^2 + 3*x*A(x)^3 ).

Links

Crossrefs

Cf. A356782, A117940, A370546.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A=concat(A,0);
F=Ser(A); A[#A] = polcoeff(subst(F,x,x*F^2 + 3*x*F^3) - F^3,#A+1) );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) A(x)^3 = A( x*A(x)^2 * (1 + 3*A(x)) ).
(1.b) A(x)^9 = A( x*A(x)^8 * (1 + 3*A(x))*(1 + 3*A(x)^3) ).
(1.c) A(x)^27 = A( x*A(x)^26 * (1 + 3*A(x))*(1 + 3*A(x)^3)*(1 + 3*A(x)^9) ).
(1.d) A(x)^(3^n) = A( x*A(x)^(3^n-1) * Product_{k=0..n-1} (1 + 3*A(x)^(3^k)) ).
(2) A(x) = x * Product_{n>=0} (1 + 3*A(x)^(3^n)).
(3) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 3*x^(3^n)) ).
(4) A(x) = x * Sum_{n>=0} A117940(n) * A(x)^n, where g.f. of A117940 equals Product{k>=0} 1 + 3*x^(3^k).
a(n) ~ c * d^n / n^(3/2), where d = 6.5583689184153129045048... and c = 0.129061736750222730297... - Vaclav Kotesovec, Apr 05 2024
The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 3^(n+1) * A(r)^(3^n) / (1 + 3*A(r)^(3^n)) and r = A(r) / Product_{n>=0} (1 + 3*A(r)^(3^n)), where r = 0.1524769363297159918479... = 1/d (d is given above) and A(r) = 0.3905308673397427979651361312666180120359942797557... - Paul D. Hanna, May 22 2024

A370546 Expansion of g.f. satisfying A(x) = A( x*A(x)^4 + 5*x*A(x)^5 )^(1/5).

Original entry on oeis.org

1, 5, 25, 125, 625, 3130, 15800, 81625, 443125, 2609375, 16984500, 121023875, 914745625, 7093331250, 55129765625, 424092582500, 3212747690625, 23952422065625, 176059004593750, 1279867522656250, 9237023201350000, 66454031585359375, 478427499949687500, 3458191615224687500
Offset: 1

Views

Author

Paul D. Hanna, Mar 27 2024

Keywords

Examples

			G.f.: A(x) = x + 5*x^2 + 25*x^3 + 125*x^4 + 625*x^5 + 3130*x^6 + 15800*x^7 + 81625*x^8 + 443125*x^9 + 2609375*x^10 + 16984500*x^11 + 121023875*x^12 + ...
where A(x)^5 = A( x*A(x)^4 + 5*x*A(x)^5 ).

Links

Crossrefs

Cf. A356782, A370439, A370545.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A=concat(A,0);
F=Ser(A); A[#A] = polcoeff(subst(F,x,x*F^4 + 5*x*F^5) - F^5,#A+3) );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) A(x)^5 = A( x*A(x)^4 * (1 + 5*A(x)) ).
(1.b) A(x)^25 = A( x*A(x)^24 * (1 + 5*A(x))*(1 + 5*A(x)^5) ).
(1.c) A(x)^125 = A( x*A(x)^124 * (1 + 5*A(x))*(1 + 5*A(x)^5)*(1 + 5*A(x)^25) ).
(1.d) A(x)^(5^n) = A( x*A(x)^(5^n-1) * Product_{k=0..n-1} (1 + 5*A(x)^(5^k)) ).
(2) A(x) = x * Product_{n>=0} (1 + 5*A(x)^(5^n)).
(3) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 5*x^(5^n)) ).

A372534 Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x) + 3*x*A(x)^2 ).

Original entry on oeis.org

1, 3, 12, 63, 372, 2322, 15102, 101439, 698340, 4900914, 34931808, 252185238, 1840242546, 13551558336, 100579610790, 751610709279, 5650352546628, 42702935642082, 324256445598816, 2472613511240754, 18926918200655928, 145379893260849876, 1120198916414984148, 8656357557290045382
Offset: 1

Views

Author

Paul D. Hanna, May 29 2024

Keywords

Comments

Compare to the following identities of the Catalan function C(x) = x + C(x)^2 (A000108):
(1) C(x)^2 = C( x*C(x)*(1 + C(x)) ),
(2) C(x)^4 = C( x*C(x)^3*(1 + C(x))*(1 + C(x)^2) ),
(3) C(x)^8 = C( x*C(x)^7*(1 + C(x))*(1 + C(x)^2)*(1 + C(x)^4) ),
(4) C(x)^(2^n) = C( x*C(x)^(2^n-1)*Product_{k=0..n-1} (1 + C(x)^(2^k)) ) for n > 0.

Examples

			G.f.: A(x) = x + 3*x^2 + 12*x^3 + 63*x^4 + 372*x^5 + 2322*x^6 + 15102*x^7 + 101439*x^8 + 698340*x^9 + 4900914*x^10 + 34931808*x^11 + 252185238*x^12 + ...
where A(x)^2 = A( x*A(x) + 3*x*A(x)^2 ).
Also,
A(x) = x * (1 + 3*A(x)) * (1 + 3*A(x)^2) * (1 + 3*A(x)^4) * (1 + 3*A(x)^8) * (1 + 3*A(x)^16) * ... * (1 + 3*A(x)^(2^n)) * ...
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 33*x^4 + 198*x^5 + 1266*x^6 + 8388*x^7 + 57033*x^8 + 396090*x^9 + 2798718*x^10 + 20056824*x^11 + 145438146*x^12 + ...
x*A(x) + 3*x*A(x)^2 = x^2 + 6*x^3 + 30*x^4 + 162*x^5 + 966*x^6 + 6120*x^7 + 40266*x^8 + 272538*x^9 + 1886610*x^10 + 13297068*x^11 + ...
SPECIFIC VALUES.
A(1/9) = 0.20017482594200170883488591841314367600913783...
A(1/10) = 0.15939222988059047986391116283589184626082823...
A(1/11) = 0.13474373940944085584086064879196682498369755...
A(1/12) = 0.11741441277153705906655653078308588616286400...

Links

Crossrefs

Cf. A356782, A371716, A372530, A000120, A000108.

Programs

  • PARI
    {a(n) = my(A = serreverse( x/prod(k=0, #binary(n), 1 + 3*x^(2^k) +x*O(x^n)) ));
polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0); F=Ser(A);
A[#A] = polcoeff( subst(F, x, x*F*(1 + 3*F) ) - F^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*A(x) + 3*x*A(x)^2 ).
(2) A(x)^4 = A( x*A(x)^3*(1 + 3*A(x))*(1 + 3*A(x)^2) ).
(3) A(x)^8 = A( x*A(x)^7*(1 + 3*A(x))*(1 + 3*A(x)^2)*(1 + 3*A(x)^4) ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1)*Product_{k=0..n-1} (1 + 3*A(x)^(2^k)) ) for n > 0.
(5) A(x) = x * Product_{n>=0} (1 + 3*A(x)^(2^n)).
(6) A(x) = x * Sum_{n>=0} 3^A000120(n) * A(x)^n, where A000120(n) = number of 1's in binary expansion of n.
(7) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 3*x^(2^n)) ).
The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 3*2^n * A(r)^(2^n) / (1 + 3*A(r)^(2^n)) and r = A(r) / Product_{n>=0} (1 + 3*A(r)^(2^n)), where r = 0.121354219013536538658862726712953201279180864478537... and A(r) = 0.301069983372147236415588688159692129761365234627514...

A372578 Expansion of g.f. A(x) satisfying A( x*A(x) + 2*A(x)^3 ) = A(x)^2.

Original entry on oeis.org

1, 2, 10, 60, 406, 2940, 22304, 174960, 1407582, 11550396, 96299472, 813433712, 6946442776, 59872428672, 520174647424, 4550665293920, 40052871669422, 354421196057404, 3151211548631856, 28137903707808048, 252219507331523688, 2268719274696321856, 20472066335198022080, 185268984285773695200
Offset: 1

Views

Author

Paul D. Hanna, Jun 17 2024

Keywords

Examples

			G.f.: A(x) = x + 2*x^2 + 10*x^3 + 60*x^4 + 406*x^5 + 2940*x^6 + 22304*x^7 + 174960*x^8 + 1407582*x^9 + 11550396*x^10 + 96299472*x^11 + 813433712*x^12 + ...
where A( x*A(x) + 2*A(x)^3 ) = A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 24*x^4 + 160*x^5 + 1152*x^6 + 8704*x^7 + 68088*x^8 + 546656*x^9 + 4478720*x^10 + 37294080*x^11 + ...
A(x)^3 = x^3 + 6*x^4 + 42*x^5 + 308*x^6 + 2358*x^7 + 18612*x^8 + 150424*x^9 + 1238688*x^10 + 10355982*x^11 + 87672468*x^12 + ...
The series reversion R(x) of A(x), R(A(x)) = x, begins:
R(x) = x - 2*x^2 - 2*x^3 - 2*x^5 - 2*x^9 - 2*x^17 - 2*x^33 - 2*x^65 - 2*x^129 - 2*x^257 - 2*x^513 + ... + -2*x^(2^n+1) + ...
SPECIFIC VALUES.
A(1/10) = 0.1580645870348513671680526916072548213169829162556439...
A(1/11) = 0.1278454819475039498675733418966788971517121949516108...
A(1/12) = 0.1104694875320629136831876267359845627848091250498995...

Links

Crossrefs

Cf. A371713, A356782.

Programs

  • PARI
    {a(n) = my(A=serreverse(x - 2*x*sum(m=0,#binary(n),x^(2^m) +x*O(x^n)))); polcoeff(A,n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(A=[0,1]); for(i=1, n, A = concat(A,0); F=Ser(A); A[#A] = polcoeff( subst(F,x, x*F + 2*F^3) - F^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*A(x) + 2*A(x)^3 ).
(2) A(x)^4 = A( x*A(x)^3 + 2*A(x)^5 + 2*A(x)^6 ).
(3) A(x)^8 = A( x*A(x)^7 + 2*A(x)^9 + 2*A(x)^10 + 2*A(x)^12 ).
(4) A(x)^(2^n) = A( x*A(x)^(2^n-1) + 2*Sum_{k=0,n-1} A(x)^(2^n+2^k) ).
(5) A(x) = x + 2*Sum_{n>=0} A(x)^(2^n+1).
(6) A(x) = Series_Reversion( x - 2*x*Sum_{n>=0} x^(2^n) ).
The radius of convergence r and A(r) satisfy: 1 = Sum_{n>=0} 2*(2^n+1) * A(r)^(2^n) and r = A(r) - 2*Sum_{n>=0} A(r)^(2^n+1), where r = 0.103594274393575546296984777950632418580281502255382627835... and A(r) = 0.191573759982214348953869719011237665707785580853712880852...
Showing 1-5 of 5 results.

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