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.

A167638 Number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having no peaks at even level.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 1, 5, 5, 15, 21, 51, 85, 188, 344, 730, 1407, 2935, 5831, 12094, 24480, 50754, 103995, 216043, 446447, 930206, 1934328, 4043275, 8448882, 17716170, 37166403, 78163336, 164520540, 346935912, 732317063, 1548096255, 3275859473
Offset: 0

Views

Author

Emeric Deutsch, Nov 08 2009

Keywords

Comments

a(n) = A167637(n,0).

Examples

			a(5)=2 because we have UUUDDUUDDD and UUUUUDDDDD.

Crossrefs

Cf. A167635, A167637.

Programs

  • Maple
    G := ((1+2*z-z^3-sqrt(1-4*z^2-2*z^3+z^6))*1/2)/(z*(1+z)): Gser := series(G, z = 0, 40): seq(coeff(Gser, z, n), n = 0 .. 38);
  • Maxima
    a(n):=sum(sum(((sum((-1)^(n-j-1)*binomial(j,-k-j+i+1)*binomial(k+j-1,k-1)*binomial(2*k+j-2,k+j-1),j,0,-k+i+1))*binomial(n-i-1,n-k-i))/k,k,1,n-i),i,0,n);
/* Vladimir Kruchinin, May 06 2018 */

Formula

G.f.: (1 + 2*z - z^3 - sqrt(1 - 4*z^2 - 2*z^3 + z^6))/(2*z*(1 + z)).
a(n) = Sum_{i=0..n} Sum_{k=1..n-i} Sum_{j=0..i-k+1} (-1)^(n-j-1)*C(j,-k-j+i+1)*C(k+j-1,k-1)*C(2*k+j-2,k+j-1)*C(n-i-1,n-k-i)/k. - Vladimir Kruchinin, May 06 2018
D-finite with recurrence (n+1)*a(n) +a(n-1) +(-4*n+7)*a(n-2) +2*(-n+2)*a(n-3) +a(n-4) -a(n-5) +(n-7)*a(n-6)=0. - R. J. Mathar, Jul 24 2022
Conjecture: g.f. A(x) = 1 + (x^3)*exp(Sum_{n >= 1} g(n, x)*x^(2*n)/n), where g(n, x) = Sum_{k = 0..n} binomial(n, k)^2*(1 + x)^k. Cf. A129509. - Peter Bala, Sep 10 2024

A129507 G.f.: (1+x+x^2-sqrt(1+2x+3x^2-2x^3+x^4))*2.

Original entry on oeis.org

0, 0, 0, 4, -4, 0, 8, -16, 12, 20, -80, 116, -4, -376, 884, -764, -1308, 5816, -9128, 648, 32008, -79608, 73100, 122020, -574044, 937456, -97748, -3366892, 8659492, -8276056, -13301640, 65261640, -109501476, 14857740, 395316672, -1042421076, 1028105156, 1582877768, -8052459580
Offset: 0

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Author

Paul Barry, Apr 18 2007

Keywords

Comments

Expansion related to the asymptotic mean of the mean square error of a wireless channel.
On page 11 of Tulino and Verdu is equation (1.17) F(x,z) = (sqrt(x(1+sqrt(z))^2+1) - sqrt(x(1-sqrt(z))^2+1))^2. G.f. is F(x,x). - Michael Somos, Mar 18 2014

Examples

			G.f. = 4*x^3 - 4*x^4 + 8*x^6 - 16*x^7 + 12*x^8 + 20*x^9 - 80*x^10 + 116*x^11 + ...

Links

Crossrefs

Cf. A129509.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2 (1 + x + x^2 - Sqrt[1 + 2*x + 3*x^2 - 2*x^3 + x^4]), {x, 0, n}]; (* Michael Somos, Mar 18 2014 *)
  • PARI
    x='x+O('x^66); Vec( 2+2*x+2*x^2-2*sqrt(1+2*x+3*x^2-2*x^3+x^4) ) \\ Joerg Arndt, Jul 17 2013

Formula

G.f.: (sqrt(x(1+sqrt(x))^2+1)-sqrt(x(1-sqrt(x))^2+1))^2.
G.f.: 2+2x+2x^2-2*sqrt(1+2x+3x^2-2x^3+x^4).
D-finite with recurrence: n*a(n) +(2*n-3)*a(n-1) +3*(n-3)*a(n-2) +(9-2*n)*a(n-3) +(n-6)*a(n-4)=0 if n>5. - R. J. Mathar, Nov 05 2012
Conjecture: g.f.: 4*q^2*(1 - 1/G(0)) where G(k) = 1 + q/(1 + q^2 / G(k+1) ). - Joerg Arndt, Jul 17 2013
a(n) = A129509(n)*4.
G.f.: 4 * (1 + x - (1 + x / (1 + x^2 / (1 + x / (1 + x^2 / ...))))). (continued fraction convergence is three power series terms per iteration) - Michael Somos, Mar 19 2014
G.f.: 4*x * (1 - 1 / (1 - x + x^2 + x / (1 - x + x^2 + x / ...))). (continued fraction convergence is one power series term per iteration) - Michael Somos, Mar 18 2014
0 = a(n)*(a(n+1) -5*a(n+2) +12*a(n+3) +11*a(n+4) +7*a(n+5)) + a(n+1)*(a(n+1) -2*a(n+2) -22*a(n+3) -21*a(n+4) -11*a(n+5)) + a(n+2)*(3*a(n+2) +17*a(n+3) +22*a(n+4) +12*a(n+5)) + a(n+3)*(-3*a(n+3) -2*a(n+4) +5*a(n+5)) + a(n+4)*(-a(n+4) +a(n+5)) if n>1. - Michael Somos, Mar 18 2014

A239466 Expansion of (1 - x + x^2 + sqrt(1 + 2*x + 3*x^2 - 2*x^3 + x^4)) / 2 in powers of x.

Original entry on oeis.org

1, 0, 1, -1, 1, 0, -2, 4, -3, -5, 20, -29, 1, 94, -221, 191, 327, -1454, 2282, -162, -8002, 19902, -18275, -30505, 143511, -234364, 24437, 841723, -2164873, 2069014, 3325410, -16315410, 27375369, -3714435, -98829168, 260605269, -257026289, -395719442
Offset: 0

Views

Author

Michael Somos, Mar 19 2014

Keywords

Examples

			G.f. = 1 + x^2 - x^3 + x^4 - 2*x^6 + 4*x^7 - 3*x^8 - 5*x^9 + 20*x^10 + ...

Links

Crossrefs

Cf. A112299, A129509.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-x+x^2 +Sqrt(1+2*x+3*x^2-2*x^3+x^4))/2)); // G. C. Greubel, Aug 08 2018
  • Mathematica
    CoefficientList[Series[(1-x+x^2 +Sqrt[1+2*x+3*x^2-2*x^3+x^4])/2, {x, 0, 50}], x] (* G. C. Greubel, Aug 08 2018 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( (1 - x + x^2 + sqrt(1 + 2*x + 3*x^2 - 2*x^3 + x^4 + x * O(x^n))) / 2, n))};
  • PARI
    {a(n) = my(A = 1 + O(x)); for(k=1, ceil(n / 3), A = 1 + x^2 / (1 + x / A)); polcoeff(A, n)};

Formula

G.f.: 1 - x + x^2 + x / (1 - x + x^2 + x / (1 - x + x^2 + x / ...)). (continued fraction convergence is one power series term per iteration).
G.f.: 1 + x^2 / (1 + x / (1 + x^2 / (1 + x / ...))). (continued fraction convergence is three power series terms per iteration).
a(n) = - A129509(n) if n>2.
HANKEL transform is period 8 sequence A112299(n+5) = [1, 1, -1, 0, 1, -1, -1, 0, ...].
HANKEL transform of a(n+1) is period 8 sequence -A112299(n+4) = [0, -1, -1, 1, 0, -1, 1, 1, ...].
D-finite with recurrence: n*a(n) +(2*n-3)*a(n-1) +3*(n-3)*a(n-2) +(-2*n+9)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Jan 25 2020

A296202 Expansion of 1/(1 + x/(1 + x/(1 + x^2/(1 + x/(1 + x^3/(1 + x/(1 + x^4/(1 + ...)))))))), a continued fraction.

Original entry on oeis.org

1, -1, 2, -4, 7, -11, 16, -22, 28, -30, 18, 29, -152, 427, -988, 2060, -4002, 7354, -12868, 21472, -34054, 50838, -69920, 84186, -75275, 2395, 217417, -742554, 1860191, -4067099, 8183154, -15493168, 27886577, -47905049, 78485095, -121944988, 177329498, -234464309, 261801461, -183121605, -164852147
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 07 2017

Keywords

Crossrefs

Cf. A007325, A092869, A129509, A292799, A292854, A296201.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 + ContinuedFractionK[x^(1 + k (1 + (-1)^k)/4), 1, {k, 0, nmax}]), {x, 0, nmax}], x]
Showing 1-4 of 4 results.

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