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.

A178694 Numerators of coefficients of Maclaurin series for (1-x-x^2)^(-1/2).

Original entry on oeis.org

1, 1, 7, 17, 203, 583, 3491, 10481, 254963, 779723, 4798681, 14831831, 184091359, 573076579, 3577974043, 11196388273, 561766479043, 1764905611763, 11107979665181, 35007455563451, 441899444305669, 1396202999849369
Offset: 0

Views

Author

Clark Kimberling, Jun 04 2010

Keywords

Comments

a(n) is also the numerator of I^(-n)*P_{n}(I/2) with I^2=-1 and P_{n} is the Legendre polynomial of degree n. - Alyssa Byrnes and C. Vignat, Jan 31 2013

Examples

			The Maclaurin series begins with 1 + (1/2)x + (7/8)x^2 + (17/16)x^3.

Links

Crossrefs

Cf. A178693.
Cf. A046161 (denominators).

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/Sqrt(1-x-x^2) )); [Numerator(Factorial(n-1)*b[n]): n in [1..m]]; // G. C. Greubel, Jan 25 2019
  • Mathematica
    Numerator[CoefficientList[Series[(1-x-x^2)^(-1/2),{x,0,30}],x]] (* Harvey P. Dale, Oct 02 2012 *)
Table[Numerator[I^(-n)*LegendreP[n, I/2]], {n, 0, 30}] (* Alyssa Byrnes and C. Vignat, Jan 31 2013 *)
  • Maxima
    b[n]:=if n<2 then 1/2^n else (1-1/n/2)*b[n-1]+(1-1/n)*b[n-2]$
a[n]:=num(b[n])$
makelist(a[n],n,0,50); /* Tani Akinari, Sep 14 2023 */
  • PARI
    a(n)=numerator(I^-n*pollegendre(n,I/2)) \\ Charles R Greathouse IV, Mar 18 2017

Formula

G.f.: (1-x-x^2)^(-1/2) (of the series, not of this sequence).
G.f.: 1/sqrt(1-x-x^2) = G(0), where G(k)= 1 + x*(1+x)*(4*k+1)/( 4*k+2 - x*(1+x)*(4*k+2)*(4*k+3)/(x*(1+x)*(4*k+3) + 4*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 08 2013
a(n) = numerator(b(n)), where b(n) = (1-1/n/2)*b(n-1)+(1-1/n)*b(n-2), with b(0)=1 and b(1)=1/2. - Tani Akinari, Sep 14 2023
a(n) = numerator(1/2^n*hypergeom([-n/2,(1-n)/2],[1],5)). - Gerry Martens, Sep 24 2023

A179191 Expansion of o.g.f. (1/2)*(-1 + 1/sqrt(1 - 4*x - 4*x^2)).

Original entry on oeis.org

0, 1, 4, 16, 68, 296, 1312, 5888, 26672, 121696, 558464, 2574848, 11917952, 55345408, 257741824, 1203224576, 5629027072, 26383656448, 123868321792, 582414688256, 2742116907008, 12926036258816, 60998951747584, 288147689046016, 1362407763795968, 6447125560016896
Offset: 0

Views

Author

Clark Kimberling, Jul 01 2010

Keywords

Comments

G.f. A(x) satisfies A(x)^2 + A(x) = (x^2 + x)/(1 - 4*x - 4*x^2). - Michael Somos, Jan 28 2019

Examples

			G.f. = x + 4*x^2 + 16*x^3 + 68*x^4 + 296*x^5 + 1312*x^6 + 5888*x^7 + ....

Links

Crossrefs

Cf. A178693, A178694, A179190.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (-1 + 1/Sqrt(1-4*x-4*x^2))/2 )); // G. C. Greubel, Jan 25 2019
  • Mathematica
    CoefficientList[1/2 (-1 + (1-4x-4x^2)^(-1/2)) + O[x]^23, x] (* Jean-François Alcover, Jul 27 2018 *)
  • Maxima
    a(n):=sum(m*sum(sum(binomial(i-1,k-1)*binomial(i,n-i),i,k,n)*sum(binomial(j,2*j-m-k)*binomial(k,j),j,0,k)/k,k,m,n),m,1,n); /* Vladimir Kruchinin, Mar 11 2011 */
a(n):=sum(2^(n-k-1)*binomial(n,k)*binomial(n-k,k),k,0,n); /* Vladimir Kruchinin, Mar 12 2015 */
  • PARI
    my(x='x+O('x^30)); concat([0], Vec((-1 +1/sqrt(1-4*x-4*x^2))/2)) \\ G. C. Greubel, Jan 25 2019
  • Sage
    ((-1 + 1/sqrt(1-4*x-4*x^2))/2).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Jan 25 2019

Formula

G.f.: (1/2)*(-1 + 1/sqrt(1 - 4*x - 4*x^2)).
G.f.: A(x) = x*A001006(A000045(x)/x-1)/(1-x*A001006(A000045(x)/x-1)).
a(n) = Sum_{m=1..n} m*Sum_{k=m..n} (Sum_{i=k..n} binomial(i-1,k-1)*binomial(i,n-i))*(Sum_{j=0..k} binomial(j,2*j-m-k)*binomial(k,j))/k. - Vladimir Kruchinin, Mar 11 2011
a(n) = Sum_{k=0..n} 2^(n-k-1)*binomial(n,k)*binomial(n-k,k). - Vladimir Kruchinin, Mar 12 2015
From Vaclav Kotesovec, Jan 26 2019: (Start)
D-finite with recurrence: n*a(n) = 2*(2*n - 1)*a(n-1) + 4*(n-1)*a(n-2).
a(n) ~ 2^(n - 7/4) * (1 + sqrt(2))^(n + 1/2) / sqrt(Pi*n). (End)
0 = a(n)*(16*a(n+1) +24*a(n+2) -8*a(n+3)) + a(n+1)*(+8*a(n+1) +16*a(n+2) -6*a(n+3)) + a(n+2)*(-2*a(n+2) +a(n+3)) for all n in Z except n=-1. - Michael Somos, Jan 27 2019
a(n) = A006139(n)/2, n>0. - R. J. Mathar, Jan 24 2020

A179190 Coefficient [x^n] of the Maclaurin series for 2 - sqrt(1 - 4*x - 4*x^2).

Original entry on oeis.org

1, 2, 4, 8, 24, 80, 288, 1088, 4256, 17088, 70016, 291584, 1230592, 5251584, 22623232, 98248704, 429677056, 1890700288, 8364824576, 37186449408, 166030266368, 744180244480, 3347321831424, 15104525959168, 68357598756864
Offset: 0

Views

Author

Clark Kimberling, Jul 01 2010

Keywords

Examples

			The Maclaurin series is 1 + 2*x + 4*x^2 + 8*x^3 + 24*x^4 + ...

Links

Crossrefs

Cf. A178693, A178694, A179191.

Programs

  • Maple
    A179190 := proc(n) if n = 0 then 1; else add( doublefactorial(2*n-2*k-3) *2^(n-k) / k! / (n-2*k)!, k=0..floor(n/2)) ; end if; end proc: # R. J. Mathar, Jul 11 2011
  • Mathematica
    Table[SeriesCoefficient[Series[2-Sqrt[1-4*t-4*t^2], {t,0,n}], n], {n, 0, 30}] (* G. C. Greubel, Jan 25 2019 *)
  • Maxima
    makelist(coeff(taylor(2-sqrt(1-4*x-4*x^2), x, 0, n), x, n), n, 0, 24); /* Bruno Berselli, Jul 04 2011 */

Formula

G.f.: 2 - sqrt(1 - 4*x - 4*x^2).
a(n) = 4*A071356(n-2), n >= 2. - R. J. Mathar, Jul 08 2010
a(n) = Sum_{k=0..floor(n/2)} (2*n - 2*k - 3)!! *2^(n-k)/(k!*(n-2k)!), n > 0. - R. J. Mathar, Jul 11 2011
a(n) ~ 2^(n - 1/4) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jan 26 2019
D-finite with recurrence: n*a(n) +2*(-2*n+3)*a(n-1) +4*(-n+3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020
Showing 1-3 of 3 results.

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