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.

A255688 G.f.: (2*x+1)/(2*sqrt(4*x^2-8*x+1)) + 1/2.

Original entry on oeis.org

1, 3, 15, 90, 579, 3858, 26262, 181380, 1265955, 8906706, 63058530, 448716876, 3206387790, 22992276180, 165364807308, 1192393813320, 8617219956003, 62397513984210, 452607991376490, 3288138397237884, 23921128800374874, 174244027232971548, 1270655996584434420
Offset: 0

Views

Author

Vladimir Kruchinin, Mar 15 2015

Keywords

Examples

			G.f. = 1 + 3*x + 15*x^2 + 90*x^3 + 579*x^4 + 3858*x^5 + 26262*x^6 + ... - _Michael Somos_, May 26 2022

Links

Crossrefs

Cf. A098663, A110170.

Programs

  • Mathematica
    CoefficientList[Series[(2*x+1)/(2*Sqrt[4*x^2-8*x+1])+1/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 15 2015 *)
  • Maxima
    a(n):=sum(2^(n-i)*binomial(n,i)*binomial(n+i-1,i),i,0,n);
  • PARI
    my(x='x+O('x^50)); Vec((2*x+1)/(2*sqrt(4*x^2-8*x+1)) + 1/2) \\ G. C. Greubel, Jun 03 2017
  • Sage
    a = lambda n: 2^n*hypergeometric([-n, n], [1], -1/2).simplify()
[a(n) for n in range(21)] # Peter Luschny, Mar 15 2015

Formula

a(n) = Sum_{i=0..n} 2^(n-i)*binomial(n,i)*binomial(n+i-1,i).
a(n) ~ 3^(1/4) * 2^(n-1) * (2+sqrt(3))^n / sqrt(Pi*n). - Vaclav Kotesovec, Mar 15 2015
a(n) = 2^n*hypergeom([-n, n], [1], -1/2). - Peter Luschny, Mar 15 2015
D-finite with recurrence: n*a(n) -6*n*a(n-1) +12*(-n+3)*a(n-2) +8*(n-3)*a(n-3)=0. - R. J. Mathar, Jan 25 2020
From Michael Somos, May 26 2022: (Start)
a(n) = (-2)^n * (P_n(-2) - P_{n-1}(-2))/2 if n>0 where P_n(x) is Legendre polynomial.
0 = a(n)*(+64*a(n+1) -96*a(n+2) -192*a(n+3) +32*a(n+4)) +a(n+1)*(-96*a(n+1) +240*a(n+2) +272*a(n+3) -48*a(n+4)) +a(n+2)*(-144*a(n+2) +60*a(n+3) -6*a(n+4)) + a(n+3)*(-6*a(n+3) +a(n+4)) if n>=0. (End)
From Peter Bala, Nov 08 2022: (Start)
a(n) = [x^n] ( (1 + 2*x)/(1 - x) )^n.
The Gauss congruences hold: a(n*p^r) == a(n^p^(r-1)) (mod p^r) for all primes p and all positive integers n and r. (End)
a(n) = 3*A098663(n-1) = Sum_{k = 0..n-1} binomial(n-1, k)*binomial(n, k+1)* 3^(k+1) for n >= 1. - Peter Bala, Sep 02 2024

A098662 E.g.f. BesselI(0,2*sqrt(3)*x) + BesselI(1,2*sqrt(3)*x)/sqrt(3).

Original entry on oeis.org

1, 1, 6, 9, 54, 90, 540, 945, 5670, 10206, 61236, 112266, 673596, 1250964, 7505784, 14073345, 84440070, 159497910, 956987460, 1818276174, 10909657044, 20827527084, 124965162504, 239516561466, 1437099368796, 2763652632300
Offset: 0

Views

Author

Paul Barry, Sep 20 2004

Keywords

Comments

Fourth binomial transform is A098663.

Links

Programs

  • Maple
    seq(binomial(n, floor(n/2))*3^floor(n/2),n=0..30); # Robert Israel, Aug 23 2019
  • Mathematica
    With[{nn=30},CoefficientList[Series[BesselI[0,2Sqrt[3]x]+ BesselI[1, 2Sqrt[3]x]/ Sqrt[3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 01 2013 *)

Formula

G.f.: 1/sqrt(1-12*x^2) + (1-sqrt(1-12*x^2))/(6*x*sqrt(1-12*x^2));
G.f.: (1 + 6*x - sqrt(1-12*x^2))/(6*x*sqrt(1-12*x^2));
a(n) = binomial(n, floor(n/2))*3^floor(n/2).
Conjecture: (n+1)*a(n) + 6(n-1)*a(n-1) - 12n*a(n-2) + 72*(2-n)*a(n-3) = 0. - R. J. Mathar, Dec 08 2011
Conjecture confirmed using the differential equation x*(6x+1)*(12*x^2-1) * g'(x) + (6*x-1)*(12*x^2+6*x+1)*g(x) + 2*x + 1 = 0 satisfied by the g.f. - Robert Israel, Aug 23 2019

A363570 Expansion of (1 / sqrt(1 - 8*x + 4*x^2) - 1 / (1 - x)) / 3.

Original entry on oeis.org

0, 1, 7, 45, 295, 1981, 13545, 93829, 656311, 4625181, 32788657, 233567269, 1670457321, 11987269477, 86268665917, 622391877045, 4500029549911, 32598283556317, 236542093805025, 1719008077215205, 12509403045819505, 91143878730342021, 664816240262272237
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 17 2023

Keywords

Crossrefs

Cf. A030662, A047665, A069835, A098663, A363571.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[(1/Sqrt[1 - 8 x + 4 x^2] - 1/(1 - x))/3, {x, 0, nmax}], x]
nmax = 22; CoefficientList[Series[Exp[x] (Exp[3 x] BesselI[0, 2 Sqrt[3] x] - 1)/3, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 3^(k - 1), {k, 1, n}], {n, 0, 22}]
Table[(2^n LegendreP[n, 2] - 1)/3, {n, 0, 22}]

Formula

E.g.f.: exp(x) * (exp(3*x) * BesselI(0,2*sqrt(3)*x) - 1) / 3.
a(n) = Sum_{k=1..n} binomial(n,k)^2 * 3^(k-1).
a(n) = (2^n * LegendreP(n,2) - 1) / 3.
a(n) = (A069835(n) - 1) / 3.
Showing 1-3 of 3 results.

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