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.

A085614 Number of elementary arches of size n.

Original entry on oeis.org

1, 3, 16, 105, 768, 6006, 49152, 415701, 3604480, 31870410, 286261248, 2604681690, 23957864448, 222399744300, 2080911654912, 19604537460045, 185813170126848, 1770558814528770, 16951376923852800, 162984598242674670
Offset: 1

Views

Author

N. J. A. Sloane, Jul 10 2003

Keywords

Links

Crossrefs

Cf. A143018, A206300.

Programs

  • Maple
    with(combstruct); ar := {EA = Union(Sequence(EA, card >= 2), Prod(Z, Sequence(EA), Sequence(EA))), C=Union(Z, Prod(Z,Z,Sequence(EA), Sequence(EA), Sequence(Union(Sequence(EA,card>=1), Prod(Z,Sequence(EA),Sequence(EA))))))}; seq(count([EA,ar], size=i),i=1..20);
  • Mathematica
    Rest[CoefficientList[Series[1/6*Sqrt[3]*Sin[1/3*ArcSin[6*Sqrt[3]*x]] - 1/2*Cos[1/3*ArcSin[6*Sqrt[3]*x]],{x,0,20}],x]] (* Vaclav Kotesovec, Oct 21 2012 *)
Rest[CoefficientList[InverseSeries[Series[x - 3*x^2 + 2*x^3, {x, 0, 20}], x],x]] (* Vaclav Kotesovec, Aug 22 2017 *)
(* From Dixon J. Jones, Apr 15 2021: (Start) *)
Table[4^n Gamma[(3n + 2)/2]/(Gamma[(n + 2)/2](n + 1)!), {n, 0, 20}]
Table[4^n Pochhammer[(n + 2)/2, n]/(n + 1)!, {n, 0, 20}] (* End *)
  • PARI
    a(n)=if(n<1,0,polcoeff(serreverse(x-3*x^2+2*x^3+x*O(x^n)),n))

Formula

G.f. is the series reversion of x-3*x^2+2*x^3.
a(n) = 2^n*(3*n)!!/((n+1)!*n!!). - Maxim Krikun (krikun(AT)iecn.u-nancy.fr), May 25 2007
G.f.: 1/6*sqrt(3)*sin(1/3*arcsin(6*sqrt(3)*x))-1/2*cos(1/3*arcsin(6*sqrt(3)*x)). - Vaclav Kotesovec, Oct 21 2012
Conjecture: n*(n-1)*a(n) +(n-1)*(n-2)*a(n-1) -12*(3*n-5)*(3*n-7)*a(n-2) -12*(3*n-8)*(3*n-10)*a(n-3) = 0. - R. J. Mathar, Oct 18 2013
a(n) ~ 2^(n - 3/2) * 3^(3*n/2 - 1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 22 2017
From Dixon J. Jones, Apr 15 2021: (Start)
a(n) = A206300(n)/2 = abs(A224884(n))/2 for n>=1.
a(n) = 4^n Gamma((3*n + 2)/2)/(Gamma((n + 2)/2)*(n + 1)!).
a(n) = (4^n*((n + 2)/2)_n)/(n + 1)!, where (x)_k is the Pochhammer symbol. (End)

A224884 Expansion of x / Series_Reversion(x*sqrt(1 + 4*x)).

Original entry on oeis.org

1, 2, -6, 32, -210, 1536, -12012, 98304, -831402, 7208960, -63740820, 572522496, -5209363380, 47915728896, -444799488600, 4161823309824, -39209074920090, 371626340253696, -3541117629057540, 33902753847705600, -325969196485349340, 3146175557067079680, -30471769822097981160
Offset: 0

Views

Author

Paul D. Hanna, Aug 21 2013

Keywords

Comments

Signed version of A206300. - Peter Bala, Mar 05 2020

Examples

			G.f.: A(x) = 1 + 2*x - 6*x^2 + 32*x^3 - 210*x^4 + 1536*x^5 - 12012*x^6 + ..
The coefficients in the powers A(x)^n of the g.f. begin:
n= 1: [1,  2,  -6,   32,  -210,  1536,-12012,  98304, -831402, ...];
n= 2: [1,  4,  -8,   40,  -256,  1848,-14336, 116688, -983040, ...];
n= 3: [1,  6,  -6,   32,  -210,  1536,-12012,  98304, -831402, ...];
n= 4: [1,  8,   0,   16,  -128,  1008, -8192,  68640, -589824, ...];
n= 5: [1, 10,  10,    0,   -50,   512, -4620,  40960, -364650, ...];
n= 6: [1, 12,  24,   -8,     0,   168, -2048,  20592, -196608, ...];
n= 7: [1, 14,  42,    0,    14,     0,  -588,   8192,  -90090, ...];
n= 8: [1, 16,  64,   32,     0,   -32,     0,   2112,  -32768, ...];
n= 9: [1, 18,  90,   96,   -18,     0,    84,      0,   -7722, ...];
n=10: [1, 20, 120,  200,     0,    24,     0,   -240,       0, ...];
n=11: [1, 22, 154,  352,   110,     0,   -44,      0,     726, ...];
n=12: [1, 24, 192,  560,   384,   -48,     0,     96,       0, ...];
n=13: [1, 26, 234,  832,   910,     0,    52,      0,    -234, ...];
n=14: [1, 28, 280, 1176,  1792,   392,     0,    -80,       0, ...];
n=15: [1, 30, 330, 1600,  3150,  1536,  -140,      0,     150, ...];
n=16: [1, 32, 384, 2112,  5120,  4032,     0,    128,       0, ...];
n=17: [1, 34, 442, 2720,  7854,  8704,  1428,      0,    -170, ...];
n=18: [1, 36, 504, 3432, 11520, 16632,  6144,   -432,       0, ...];
n=19: [1, 38, 570, 4256, 16302, 29184, 17556,      0,     342, ...];
n=20: [1, 40, 640, 5200, 22400, 48048, 40960,   5280,       0, ...]; ...
which illustrates the property [x^n] A(x)^(n+2*k) = 0 for k=1..n-1:
[x^2] A(x)^4 = 0;
[x^3] A(x)^5 = 0, [x^3] A(x)^7 = 0;
[x^4] A(x)^6 = 0, [x^4] A(x)^8 = 0, [x^4] A(x)^10 = 0; ...
[x^5] A(x)^7 = 0, [x^5] A(x)^9 = 0, [x^5] A(x)^11 = 0, [x^5] A(x)^13 = 0; ...
Related series:
sqrt(1+4*x) = 1 + 2*x - 2*x^2 + 4*x^3 - 10*x^4 + 28*x^5 - 84*x^6 + 264*x^7 - 858*x^8 + ... + (-1)^(n-1)*2*A000108(n-1)*x^n + ...

Links

Crossrefs

Cf. A000108, A206300.
Cf. A000984, A048990, A078531, A182122, A245112, A245113.

Programs

  • Mathematica
    CoefficientList[Series[x/InverseSeries[Series[x*Sqrt[1+4*x],{x,0,20}],x],{x,0,20}],x] (* Vaclav Kotesovec, Aug 22 2013 *)
  • PARI
    {a(n)=polcoeff(x/serreverse(x*sqrt(1+4*x +x^2*O(x^n))),n)}
for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x) = A(x)^3 - 4*x.
(2) A(x) = sqrt(1 + 4*x/A(x)).
(3) A(x*sqrt(1+4*x)) = sqrt(1+4*x).
(4) [x^n] A(x)^(n+2*k) = 0 for k=1..n-1, for n >= 2.
From Vaclav Kotesovec, Aug 22 2013: (Start)
a(n) = (-1)^(n+1) * 3^(3*n/2-1) * 4^(n-1) * GAMMA(n/2 - 1/6) * GAMMA(n/2 + 1/6)/(Pi*n!).
|a(n)| ~ 6^(n-1)*3^(n/2)/(sqrt(Pi/2)*n^(3/2)).
D-finite with recurrence: (n-1)*n*a(n) = 12*(3*n-7)*(3*n-5)*a(n-2). (End)
G.f.: (2/sqrt(3))*cosh(1/3*arccosh(sqrt(108)*x)). - Vladimir Kruchinin, Oct 11 2022
G.f. A(x) satisfies A(x) = 1/A(-x/A(x)^4). - Seiichi Manyama, Jun 20 2025

A343446 Coefficients of the series S(p, q) for which -(p^(1/3))*S converges to the largest real root of x^4 - p*x + q, where 0 < p and 0 < q < 3*(p/4)^(4/3).

Original entry on oeis.org

-1, 1, 4, 40, 648, 14560, 418880, 14696640, 608608000, 29056867840, 1571364748800, 94937979136000, 6337884013260800, 463301182536192000, 36806315255277568000, 3157533815406530560000, 290912372128665391104000, 28648563542097847828480000
Offset: 0

Views

Author

Dixon J. Jones, May 26 2021

Keywords

Comments

Based on formulas for series solutions of trinomials given in Eagle article.
S(p, q) = Sum_{n>=0} (a(n)*q^n)/((3^n)*(p^(4n/3))*n!)
In general, given m > 1, p > 0 and 0 < q < m*(p/(m + 1))^((m + 1)/m), the series S(m, p, q) for which (-p^(1/m))*S converges to the largest real root of x^(m + 1) - p*x + q has coefficients c(n) = m^(n - 1)*((n + m - 1)/m)(n - 1), where (x)_k is the Pochhammer symbol for Gamma(x + k)/Gamma(k), and S(m, p, q) = Sum{n>=0}(c(n)*q^n)/((m^n)*(p^(n*(m + 1)/m)*n!).

Links

Crossrefs

A343445 relates similarly to the largest real root of x^3 - p*x + q.
A206300 relates similarly to the largest real root of x^3 - 3*u*x + 4*u, u >= 4.

Programs

  • Maple
    a := proc(n) option remember; if n < 3 then [-1, 1, 4][n+1] else 4*(4*n - 7)*(4*n - 10)*(4*n - 13)*a(n-3) fi; end:
seq(a(n), n = 0..20); # Peter Bala, Jul 23 2024
  • Mathematica
    Clear[a]; a=Table[3^(n - 1) Pochhammer[(n + 2)/3, n - 1], {n, 0, 20}]
(* In general, for the series S(m, p, q) for which (-p^(1/m))*S converges to the largest real root of x^(m + 1) - p*x + q, the first n + 1 coefficients are: *)
Clear[c]; c[m_,n_] := Table[m^(k - 1) Pochhammer[(k + m - 1)/m, k - 1], {k, 0, n}](* and S(m, p, q) to n + 1 terms is given by *)
Clear[s]; s[m_,p_,q_,n_]:= Sum[c[m,n][[k + 1]]*q^k/((m^k)*(p^(k (m + 1)/m))*k!), {k, 0, n}]

Formula

a(n) = 3^(n - 1)*((n + 2)/3)_(n - 1), where (x)_k is the Pochhammer symbol for Gamma(x + k)/Gamma(x).
a(n) = 4*(4*n - 7)*(4*n - 10)*(4*n - 13)*a(n-3) with a(0) = -1, a(1) = 1 and a(2) = 4. - Peter Bala, Jul 23 2024
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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