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.

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A088854 a(n) = (2^(n-1))*(Integral_{x=0..1} (1+x^2)^n dx)/(Integral_{x=0..1} (1-x^2)^n dx).

Original entry on oeis.org

2, 7, 24, 83, 292, 1046, 3808, 14051, 52412, 197202, 747120, 2846318, 10892936, 41844172, 161247104, 623034403, 2412871916, 9363311482, 36399254864, 141721774138, 552572485496, 2157194452852, 8431104269504, 32986010380558, 129177323979992, 506313914434036, 1986097541692128
Offset: 1

Views

Author

Al Hakanson (hawkuu(AT)excite.com), Nov 24 2003

Keywords

Examples

			a(3) = 24.

Links

Crossrefs

Cf. A082590.

Programs

  • Maple
    A088854 := n -> 2^(n-1)*JacobiP(n, 1/2, -1 - n, 3):
seq(simplify(A088854(n)), n = 1..26);  # Peter Luschny, Jan 22 2025
  • Mathematica
    f[n_] := 2^(n - 1)Integrate[(1 + x^2)^n, {x, 0, 1}] / Integrate[(1 - x^2)^n, {x, 0, 1}]; Table[ f[n], {n, 1, 24}] (* Robert G. Wilson v, Feb 27 2004 *)
Table[2^(n-1)+Sum[2^(n-k)*Binomial[2*k-1,k], {k,1,n}],{n,1,20}] (* Vaclav Kotesovec, Oct 28 2012 *)
  • PARI
    x='x+O('x^66); Vec(-1/2+1/(2*(1-2*x)*sqrt(1-4*x))) \\ Joerg Arndt, May 10 2013

Formula

G.f.: -1/2 + 1/(2*(1-2*x)*sqrt(1-4*x)). - Vladeta Jovovic, Dec 14 2003
Recurrence: n*a(n) = 2*(3*n-1)*a(n-1) - 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 14 2012
a(n) ~ 4^n/sqrt(Pi*n). - Vaclav Kotesovec, Oct 14 2012
a(n) = 2^(n-1) + Sum_{k=1..n} 2^(n-k)*C(2*k-1,k). - Vaclav Kotesovec, Oct 28 2012
2*a(n) = Sum_{k=0..n} C(2k,k)*C(2(n-k),n-k)/C(n,k). - Zhi-Wei Sun, Oct 14 2019
a(n) = 2^(n-1)*JacobiP(n, 1/2, -1 - n, 3). - Peter Luschny, Jan 22 2025

Extensions

More terms from Robert G. Wilson v, Feb 27 2004

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

A096466 Triangle T(n,k), read by rows, formed by setting all entries in the zeroth column ((n,0) entries) and the main diagonal ((n,n) entries) to powers of 2 with all other entries formed by the recursion T(n,k) = T(n-1,k) + T(n,k-1).

Original entry on oeis.org

1, 2, 2, 4, 6, 4, 8, 14, 18, 8, 16, 30, 48, 56, 16, 32, 62, 110, 166, 182, 32, 64, 126, 236, 402, 584, 616, 64, 128, 254, 490, 892, 1476, 2092, 2156, 128, 256, 510, 1000, 1892, 3368, 5460, 7616, 7744, 256, 512, 1022, 2022, 3914, 7282, 12742, 20358, 28102, 28358, 512
Offset: 0

Views

Author

Gerald McGarvey, Aug 12 2004

Keywords

Comments

T(n,k) = T(n-1,k) + T(n,k-1) for n >= 2 and 1 <= k <= n - 1 with T(n,0) = T(n,n) = 2^n for n >= 0.
The n-th row sum equals A082590(n), which is the expansion of 1/(1 - 2*x)/sqrt(1 - 4*x) and equals 2^n * JacobiP(n, 1/2, -1-n, 3).
First column is T(n,1) = A000918(n+1) = 2^(n+1) - 2.
From Petros Hadjicostas, Aug 06 2020: (Start)
T(n,2) = 2^(n+2) - 2*n - 8 for n >= 2.
T(n+1,n) = 2^n + Sum_{k=0..n} T(n,k) = 2^n + A082590(n).
Bivariate o.g.f.: ((1 - x)*(1 - y)/(1 - 2*x) - x*y/sqrt(1 - 4*x*y))/((1 - 2*x*y)*(1 - x - y)). (End)

Examples

			From _Petros Hadjicostas_, Aug 06 2020: (Start)
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
   1;
   2,   2;
   4,   6,   4;
   8,  14,  18,   8;
  16,  30,  48,  56,  16;
  32,  62, 110, 166, 182,  32;
  64, 126, 236, 402, 584, 616, 64;
  ... (End)

Crossrefs

Cf. A000918, A082590.

Programs

  • PARI
    T(n,k) = if ((k==0) || (n==k), 2^n, if ((n<0) || (k<0), 0, if (n>k, T(n-1,k) + T(n,k-1), 0)));
for(n=0, 10, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Aug 07 2020

Extensions

Offset changed to 0 by Petros Hadjicostas, Aug 06 2020

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).
Previous Showing 21-25 of 25 results.

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