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 1-5 of 5 results.

A320827 G.f.: -sqrt(1 - 4*x)*(2*x - 1)/(3*x - 1).

Original entry on oeis.org

-1, 1, 1, 3, 11, 41, 151, 549, 1977, 7075, 25229, 89831, 319881, 1140523, 4075321, 14603243, 52501659, 189440937, 686181711, 2495243373, 9109701699, 33388293177, 122840931891, 453622854873, 1681057537359, 6250742452125, 23316503569983, 87236431248445
Offset: 0

Views

Author

Peter Luschny, Oct 23 2018

Keywords

Links

Crossrefs

Cf. A002421, A029759, A320825, A320826.

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt(1-4*x)*(1-2*x)/(3*x-1))); // G. C. Greubel, Oct 27 2018
  • Maple
    ogf := x -> -sqrt(1 - 4*x)*(2*x - 1)/(3*x - 1);
ser := series(ogf(x), x, 30); seq(coeff(ser, x, k), k=0..27);
# By recurrence:
a := proc(n) option remember; if n <= 4 then return [-1,1,1,3,11][n+1] fi;
((-90+66*n-12*n^2)*a(n-2)+(30-34*n+7*n^2)*a(n-1))/((n-4)*n) end:
seq(a(n), n=0..27);
  • Mathematica
    a[n_] := (-4)^n Binomial[3/2,n]((4/3)n - 2 + Hypergeometric2F1[1,-n, 5/2 - n, 3/4]); Table[a[n], {n, 0, 27}]
CoefficientList[Series[Sqrt[1-4*x]*(1-2*x)/(3*x-1), {x, 0, 40}], x] (* G. C. Greubel, Oct 27 2018 *)
  • PARI
    x='x+O('x^40); Vec(sqrt(1-4*x)*(1-2*x)/(3*x-1)) \\ G. C. Greubel, Oct 27 2018

Formula

a(n) = (-4)^n*binomial(3/2, n)*((4/3)*n - 2 + hypergeom([1, -n], [5/2 - n], 3/4)).
D-finite with recurrence: a(n) = ((-90+66*n-12*n^2)*a(n-2) + (30-34*n+7*n^2)*a(n-1))/((n-4)*n) for n >= 5.
Expansion of -1/g.f. gives A029759.
a(n) = A320825(n) - A320826(n).

A220589 Number of simple skew-merged permutations with n elements.

Original entry on oeis.org

2, 2, 8, 16, 44, 108, 284, 740, 1966, 5254, 14172, 38476, 105122, 288754, 797036, 2209588, 6149618, 17176186, 48129284, 135261796, 381169532, 1076824852, 3049109912, 8652239496, 24600592454, 70075316198, 199955694616
Offset: 4

Views

Author

Vincent Vatter, Jan 14 2013

Keywords

Comments

A permutation is skew-merged if it is the union of an increasing subsequence and a decreasing subsequence. A permutation is simple if it does not contain a nontrivial interval.

Links

Crossrefs

Cf. A029759.

Programs

  • Mathematica
    Rest[Rest[Rest[Rest[CoefficientList[Series[(1-2*x-x^2+(x-1)*Sqrt[1-2*x-3*x^2])/(x+1), {x, 0, 20}], x]]]]] (* Vaclav Kotesovec, Jan 16 2013 *)

Formula

G.f.: (1-2*x-x^2+(x-1)*sqrt(1-2*x-3*x^2)) / (x+1).
Recurrence (for n>4): (n-4)*n*a(n) = (n^2-7*n+15)*a(n-1) + (n-3)*(5*n-17)*a(n-2) + 3*(n-4)*(n-3)*a(n-3). - Vaclav Kotesovec, Jan 16 2013
a(n) ~ 3^(n-1/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jan 16 2013

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 1-5 of 5 results.

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