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 1-8 of 8 results.

A126079 G.f.: (1-2*x)*sqrt(1-4*x).

Original entry on oeis.org

1, -4, 2, 0, -2, -8, -28, -96, -330, -1144, -4004, -14144, -50388, -180880, -653752, -2377280, -8691930, -31935960, -117858900, -436698240, -1623971580, -6059188080, -22676052360, -85100059200, -320188972740, -1207569840048, -4564276213608, -17286920538496, -65597689543400
Offset: 0

Views

Author

N. J. A. Sloane, Mar 22 2007

Keywords

Links

Crossrefs

Cf. A126966, A126089.

Programs

  • Magma
    [(3-n)*Binomial(2*n,n)/((2*n-3)*(2*n-1)): n in [0..30]]; // Vincenzo Librandi, Mar 29 2014
  • Maple
    a:=n->(3-n)*binomial(2*n,n)/(2*n-3)/(2*n-1): seq(a(n),n=0..30); # Emeric Deutsch, Mar 25 2007
A126079List := proc(m) local A, P, n; A := [1,-4,2,0]; P := [-2,0];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A126079List(27); # Peter Luschny, Mar 26 2022
  • Mathematica
    CoefficientList[Series[(1-2x)Sqrt[1-4x],{x,0,40}],x] (* Harvey P. Dale, Mar 28 2014 *)

Formula

From Emeric Deutsch, Mar 25 2007: (Start)
a(n) = binomial(2n,n) - 6*binomial(2n-2,n-1) + 8*binomial(2n-4,n-2).
a(n) = (3-n)*binomial(2n,n)/((2*n-3)*(2*n-1)). (End)
E.g.f.: 1 - 4*x + 2*x^2 - x^2*Q(0), where Q(k)= 1 - 2*x/(k+3 - (k+3)*(2*k+3)/(2*k+3 - (k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 29 2013
a(n) ~ -2^(2*n-2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 29 2013
D-finite with recurrence: n*a(n) +2*(-3*n+5)*a(n-1) +4*(2*n-7)*a(n-2)=0. - R. J. Mathar, Jan 23 2020

A126967 Expansion of e.g.f.: sqrt(1+4*x)/(1+2*x).

Original entry on oeis.org

1, 0, -4, 48, -624, 9600, -175680, 3790080, -95235840, 2752081920, -90328089600, 3328103116800, -136191650918400, 6131573025177600, -301213549769932800, 16030999766605824000, -918678402394841088000, 56387623092958789632000, -3690023220507773140992000, 256425697620583349354496000
Offset: 0

Views

Author

N. J. A. Sloane, Mar 22 2007

Keywords

Comments

A row of an array that is under investigation.

Links

Crossrefs

Cf. A126966.

Programs

  • Magma
    m:=20; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Sqrt(1+4*x)/(1+2*x))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Jan 24 2020
  • Maple
    seq(coeff(series( sqrt(1+4*x)/(1+2*x), x, n+1)*n!, x, n), n = 0..20);
# G. C. Greubel, Jan 29 2020
A126967 := n -> (-2)^n*n!*JacobiP(n, -1/2, -(n+1), 3):
seq(simplify(A126967(n)), n = 0..19);  # Peter Luschny, Jan 22 2025
  • Mathematica
    nmax=20; CoefficientList[Series[Sqrt[1 + 4 x] / (1 + 2 x), {x, 0, nmax}], x] Range[0, nmax]! (* Vincenzo Librandi, Jan 24 2020 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace( sqrt(1+4*x)/(1+2*x) )) \\ G. C. Greubel, Jan 29 2020
  • Sage
    [factorial(n)*( sqrt(1+4*x)/(1+2*x) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jan 29 2020

Formula

D-finite with recurrence: a(n) +6*(n-1)*a(n-1) +4*(n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Jan 23 2020
a(n) = (-2)^n*n!*JacobiP(n, -1/2, -(n+1), 3). - Peter Luschny, Jan 22 2025

A158495 Expansion of ((1-4*x) + sqrt(1-4*x))/(2*(1-2*x)).

Original entry on oeis.org

1, -1, -3, -8, -21, -56, -154, -440, -1309, -4048, -12958, -42712, -144210, -496432, -1735764, -6145968, -21986781, -79331232, -288307254, -1054253208, -3875769606, -14315659632, -53097586284, -197677736208, -738415086066
Offset: 0

Views

Author

Paul Barry, Mar 20 2009

Keywords

Comments

Hankel transform is A158496.

Links

Crossrefs

Essentially the same as A014318, up to sign and offset.
Cf. A126966, A158496, A246432.

Programs

  • Magma
    A158495:= func< n | n eq 0 select 1 else - (&+[2^(n-j)*Catalan(j-1): j in [1..n]]) >;
[A158495(n): n in [0..40]]; // G. C. Greubel, Jan 09 2023
  • Mathematica
    CoefficientList[Series[((1-4x)+Sqrt[1-4x])/(2(1-2x)),{x,0,30}],x] (* Harvey P. Dale, Dec 15 2011 *)
  • SageMath
    def A158495(n): return int(n==0) - sum(2^(n-k)*catalan_number(k-1) for k in range(1,n+1))
[A158495(n) for n in range(41)] # G. C. Greubel, Jan 09 2023

Formula

a(n) = (2*0^n - 2^n + A126966(n))/2.
Conjecture: n*a(n) +6*(-n+1)*a(n-1) +4*(2*n-3)*a(n-2)=0. - R. J. Mathar, Dec 03 2014
From G. C. Greubel, Jan 09 2023: (Start)
a(n) = [n=0] - Sum_{k=1..n} 2^(n-k)*A000108(k-1).
a(n) = Sum_{j=0..n} 2^(n-j)*A246432(j). (End)

A134635 Row sums of triangle A134634.

Original entry on oeis.org

1, 2, 6, 18, 54, 164, 508, 1610, 5222, 17308, 58484, 200948, 700348, 2470472, 8804024, 31648858, 114623366, 417820972, 1531629764, 5642508508, 20878731476, 77561756152, 289156105544, 1081466311108, 4056621689564, 15257327887384, 57525469116168, 217383333920040, 823195469508792, 3123379468819600, 11872247508521072, 45203794091311354
Offset: 0

Views

Author

Gary W. Adamson, Nov 04 2007

Keywords

Examples

			a(3) = 18 = sum of row 3 terms of triangle A134634: (5 + 4 + 4 + 5).

Links

Crossrefs

Cf. A000108, A126966, A134634.

Programs

  • Maple
    A134635 := n -> 2*binomial(2*n, n)/(n+1) + add(2^k*binomial(2*n-2*k, n-k)/(2*n-2*k-1), k=0..n): seq(A134635(n), n=0..31); # Mélika Tebni, Feb 11 2024

Formula

Conjecture: (n+1)*a(n) +(-7*n+1)*a(n-1) +2*(7*n-8)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, May 30 2014
a(n) = 2*A000108(n) - A126966(n). - Mélika Tebni, Feb 11 2024

Extensions

Corrected and extended by N. J. A. Sloane, Feb 18 2013

A137720 Expansion of sqrt(1-4*x)/(1-3*x).

Original entry on oeis.org

1, 1, 1, -1, -13, -67, -285, -1119, -4215, -15505, -56239, -202309, -724499, -2589521, -9254363, -33111969, -118725597, -426892131, -1539965973, -5575175319, -20260052337, -73908397851, -270657727593, -994938310059
Offset: 0

Views

Author

Paul Barry, Feb 08 2008

Keywords

Comments

Hankel transform is A120617. In general, sqrt(1-4*x)/(1-k*x) has Hankel transform with g.f. of (1-2*x)/(1+2*(k+2)*x+4*x^2).

Links

Crossrefs

Cf. A126966, A106191, A157674.

Programs

  • Mathematica
    CoefficientList[Series[Sqrt[1-4*x]/(1-3*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 31 2014 *)
FullSimplify[Table[I*3^(-1/2+n) + 2^(1+2*n)*Gamma[1/2+n] * Hypergeometric2F1Regularized[1, 1/2+n, 2+n, 4/3]/(3*Sqrt[Pi]), {n, 0, 20}]] (* Vaclav Kotesovec, Jul 31 2014 *)
  • PARI
    x='x+O('x^50); Vec(sqrt(1-4*x)/(1-3*x)) \\ G. C. Greubel, Mar 21 2017

Formula

a(n) = Sum_{k=0..n} 3^k*C(2*n-2*k,n-k)/(1-(2*n-2*k)).
D-finite with recurrence: n*a(n) + (6-7*n)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Nov 16 2011
a(n) ~ -2^(2*n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 31 2014
a(n) = (-1)^n * A157674(2*n+1). - Vaclav Kotesovec, Jul 31 2014

A372239 Expansion of (1 + 2*x) / ((1 - 2*x)*sqrt(1 - 4*x)).

Original entry on oeis.org

1, 6, 22, 76, 262, 916, 3260, 11800, 43334, 161028, 604052, 2283048, 8681116, 33171144, 127260088, 489870896, 1891057222, 7317881444, 28378110628, 110251755656, 429040567732, 1672032067544, 6524678847688, 25490986350416, 99696437839132, 390298689482216
Offset: 0

Views

Author

Mélika Tebni, Apr 23 2024

Keywords

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 6 == 0 (mod p).
Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 2 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028283, A029759, A082590, A126966, A188622.

Programs

  • Maple
    a := n -> binomial(2*n,n) + 4*add(2^(n-k-1)*binomial(2*k,k), k = 0 .. n-1):
seq(a(n), n = 0 .. 25);
# Second program:
a:= proc(n) option remember; `if`(n=0,1,2*a(n-1)+2*binomial(2*n-2, n-1)*(3*n-1)/n) end: seq(a(n), n = 0 .. 25);
# Recurrence:
a := proc(n) option remember; if n < 2 then return [1, 6][n + 1] fi;
((-18*(n - 2)^2 - 42*n + 66)*a(n - 1) + 4*(3*n - 1)*(2*n - 3)*a(n - 2)) / (n*(4 - 3*n)) end: seq(a(n), n = 0..25);  # Peter Luschny, Apr 23 2024

Formula

a(n) = 5*A000984(n) - 4* A029759(n) = binomial(2*n,n) + 4*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
a(n) = 2*a(n-1) + A028283(n) = 2*a(n-1) + 2*binomial(2n-2, n-1)*(3*n-1)/n for n >= 1.
a(n) = 2*A082590(n-1) + A082590(n) for n >= 1.
a(n) = 2*A188622(n) - A126966(n).
D-finite with recurrence n*a(n) +2*(-2*n-1)*a(n-1) +4*(-n+6)*a(n-2) +8*(2*n-5)*a(n-3)=0. - R. J. Mathar, Apr 24 2024
E.g.f.: exp(2*x)*(BesselI(0, 2*x)*(1 + 4*x + 2*Pi*x*StruveL(1, 2*x)) - 2*Pi*x*BesselI(1, 2*x)*StruveL(0, 2*x)). - Stefano Spezia, Aug 29 2025

A372420 Expansion of (1 + x) / ((1 - 2*x)*sqrt(1 - 4*x)).

Original entry on oeis.org

1, 5, 18, 62, 214, 750, 2676, 9708, 35718, 132926, 499228, 1888644, 7186876, 27478508, 105474216, 406182552, 1568563014, 6071812638, 23552366796, 91525132692, 356242058004, 1388588519268, 5419533876696, 21176597444712, 82834229300124, 324326668721100
Offset: 0

Views

Author

Mélika Tebni, Apr 30 2024

Keywords

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 5 == 0 (mod p).
Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 1 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028270, A029759, A082590, A126966, A188622.

Programs

  • Maple
    a := n -> -2^(n-1)*3*I + binomial(2*n, n)*(1-3/2*hypergeom([1, n+1/2], [n+1], 2)):
seq(simplify(a(n)), n = 0 .. 25);
  • PARI
    my(x='x+O('x^40)); Vec((1 + x) / ((1 - 2*x)*sqrt(1 - 4*x))) \\ Michel Marcus, Apr 30 2024

Formula

a(n) = 4*A000984(n) - 3* A029759(n) = binomial(2*n,n) + 3*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
a(n) = 2*a(n-1) + A028270(n) = 2*a(n-1) + binomial(2*n, n) + binomial(2*n-2, n-1) for n >= 1.
a(n) = - 2^(n-1)*3*i + binomial(2*n,n)*(1-3/2*hypergeom([1,n+1/2],[n + 1],2)).
a(n) = A082590(n-1) + A082590(n) for n >= 1.
a(n) = (5*A188622(n) - 2*A126966(n)) / 3.
D-finite with recurrence n*a(n) -5*n*a(n-1) +2*(n+5)*a(n-2) +4*(2*n-5)*a(n-3)=0. - R. J. Mathar, May 01 2024

A372611 Expansion of (1 + 3*x) / ((1 - 2*x)*sqrt(1 - 4*x)).

Original entry on oeis.org

1, 7, 26, 90, 310, 1082, 3844, 13892, 50950, 189130, 708876, 2677452, 10175356, 38863780, 149045960, 573559240, 2213551430, 8563950250, 33203854460, 128978378620, 501839077460, 1955475615820, 7629823818680, 29805375256120, 116558646378140, 456270710243332
Offset: 0

Views

Author

Mélika Tebni, May 07 2024

Keywords

Comments

Conjecture: For p Pythagorean prime (A002144), a(p) - 7 == 0 (mod p).
Conjecture: For p prime of the form 4*k + 3 (A002145), a(p) + 3 == 0 (mod p).

Crossrefs

Cf. A002144, A002145, A000984, A028322, A029759, A082590, A126966, A188622, A372239, A372420.

Programs

  • Maple
    a := n -> -2^(n-1)*5*I + binomial(2*n, n)*(1-5/2*hypergeom([1, n+1/2], [n+1], 2)): seq(simplify(a(n)), n = 0 .. 25);
  • PARI
    my(x='x+O('x^30)); Vec((1 + 3*x) / ((1 - 2*x)*sqrt(1 - 4*x))) \\ Michel Marcus, May 07 2024

Formula

a(n) = 6*A000984(n) - 5* A029759(n) = binomial(2*n,n) + 5*Sum_{k=0..n-1} 2^(n-k-1)*binomial(2*k,k).
a(n) = 2*a(n-1) + A028322(n) = 2*a(n-1) + binomial(2*n, n) + 3*binomial(2*n-2, n-1) for n >= 1.
a(n) = - 2^(n-1)*5*i + binomial(2*n,n)*(1-5/2*hypergeom([1, n + 1/2], [n + 1], 2)).
a(n) = 3*A082590(n-1) + A082590(n) for n >= 1.
a(n) = (7*A188622(n) - 4*A126966(n))/3.
a(n) = 2*A372239(n) - A372420(n).
Showing 1-8 of 8 results.

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