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 1-7 of 7 results.

A108488 Expansion of 1/sqrt(1 -2*x -3*x^2 -4*x^3 +4*x^4).

Original entry on oeis.org

1, 1, 3, 9, 23, 69, 203, 601, 1815, 5493, 16731, 51225, 157367, 485093, 1499499, 4646233, 14427095, 44880981, 139849979, 436419737, 1363713015, 4266417221, 13362194571, 41891406681, 131452430999, 412835452213, 1297543367835
Offset: 0

Views

Author

Paul Barry, Jun 04 2005

Keywords

Comments

In general, Sum_{k=0..n} C(n-k,k)^2*a^k*b^(n-k) has the expansion 1/sqrt(1 -2*b*x -(2*a*b -b^2)*x^2 -2*a*b^2*x^3 +(a*b)^2*x^4).
Diagonal of the rational function 1 / ((1 - x)*(1 - y) - 2*(x*y)^2). - Ilya Gutkovskiy, Apr 23 2025

Links

Crossrefs

Cf. A051286, A108480.

Programs

  • Mathematica
    Table[Sum[Binomial[n-k,k]^2*2^k,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 24 2013 *)
CoefficientList[Series[1/Sqrt[1-2x-3x^2-4x^3+4x^4],{x,0,30}],x] (* Harvey P. Dale, Apr 06 2023 *)
  • PARI
    {a(n)=polcoeff( exp(sum(m=1, n, sum(k=0, m, binomial(2*m, 2*k) * 2^k * x^k) *x^m/m) +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 31 2014

Formula

a(n) = Sum_{k=0..n} C(n-k, k)^2*2^k.
a(n) ~ ((4*sqrt(2)-1)/62)^(1/4) * (1+2*sqrt(2)+sqrt(1+4*sqrt(2)))^(n+1) /(sqrt(Pi*n)*2^(n+2)). - Vaclav Kotesovec, Jul 24 2013
D-finite with recurrence: n*a(n) +(-2*n+1)*a(n-1) +3*(-n+1)*a(n-2) +2*(-2*n+3)*a(n-3) +4*(n-2)*a(n-4)=0. - R. J. Mathar, Aug 06 2013
G.f.: exp( Sum_{n>=1} (x^n/n) * Sum_{k=0..n} C(2*n,2*k) * 2^k * x^k ). - Paul D. Hanna, Aug 31 2014

A375273 Expansion of 1/(1 - 2*x - 3*x^2 - 4*x^3 + 4*x^4).

Original entry on oeis.org

1, 2, 7, 24, 73, 238, 763, 2436, 7821, 25050, 80255, 257200, 824081, 2640582, 8461187, 27111644, 86872853, 278363058, 891946503, 2858027016, 9157854361, 29344123550, 94026132235, 301283944500, 965391362461, 3093362593162, 9911930522767, 31760378496864
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A182890, A375255.
Cf. A108480, A108488.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(1/(1-2*x-3*x^2-4*x^3+4*x^4))
  • PARI
    a(n) = sum(k=0, n\2, 2^k*binomial(2*n-2*k+2, 2*k+1))/2;

Formula

a(n) = 2*a(n-1) + 3*a(n-2) + 4*a(n-3) - 4*a(n-4).
a(n) = (1/2) * Sum_{k=0..floor(n/2)} 2^k * binomial(2*n-2*k+2,2*k+1).

A375275 Expansion of (1 - x + x^2)/(1 - 2*x + 3*x^2 + 2*x^3 + x^4).

Original entry on oeis.org

1, 1, 0, -5, -13, -12, 25, 117, 196, 3, -841, -2200, -2079, 4121, 19720, 33435, 1547, -140772, -372775, -359763, 678796, 3323203, 5702319, 437200, -23557759, -63154959, -62213360, 111716475, 559940707, 972313668, 103585625, -3941367643, -10698060204
Offset: 0

Views

Author

Seiichi Manyama, Aug 09 2024

Keywords

Links

Crossrefs

Cf. A108479, A108480, A108484.
Cf. A375255.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec((1-x+x^2)/(1-2*x+3*x^2+2*x^3+x^4))
  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-2*k, 2*k));

Formula

a(n) = 2*a(n-1) - 3*a(n-2) - 2*a(n-3) - a(n-4).
a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-2*k,2*k).

A387622 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(2*n-4*k,2*k).

Original entry on oeis.org

1, 1, 1, 3, 13, 31, 61, 151, 413, 1031, 2445, 5991, 15069, 37447, 91917, 226503, 561373, 1389735, 3431501, 8474983, 20955229, 51814407, 128054029, 316455559, 782209629, 1933537511, 4779082829, 11812031271, 29195752157, 72164132167, 178368130061
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A001541, A108480, A387623.

Programs

  • Magma
    [&+[2^k* Binomial(2*n-4*k, 2*k): k in [0..Floor (n/3)]]: n in [0..30]]; // Vincenzo Librandi, Sep 05 2025
  • Mathematica
    Table[Sum[2^k*Binomial[2*n-4*k,2*k],{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 05 2025 *)
  • PARI
    a(n) = sum(k=0, n\3, 2^k*binomial(2*n-4*k, 2*k));

Formula

G.f.: (1-x-2*x^3)/((1-x-2*x^3)^2 - 8*x^4).
a(n) = 2*a(n-1) - a(n-2) + 4*a(n-3) + 4*a(n-4) - 4*a(n-6).

A387623 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(2*n-6*k,2*k).

Original entry on oeis.org

1, 1, 1, 1, 3, 13, 31, 57, 95, 193, 463, 1081, 2295, 4609, 9423, 20185, 44071, 94801, 199807, 418921, 885879, 1889889, 4034639, 8573561, 18155399, 38461105, 81665695, 173627401, 368961431, 783201921, 1661811055, 3527298329, 7490519335, 15908549329, 33779968447
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2025

Keywords

Links

Crossrefs

Cf. A001541, A108480, A387622.
Cf. A387626, A387629.

Programs

  • Magma
    [&+[2^k* Binomial(2*n-6*k, 2*k): k in [0..Floor (n/4)]]: n in [0..30]]; // Vincenzo Librandi, Sep 05 2025
  • Mathematica
    Table[Sum[2^k*Binomial[2*n-6*k,2*k],{k,0,Floor[n/4]}],{n,0,40}] (* Vincenzo Librandi, Sep 05 2025 *)
  • PARI
    a(n) = sum(k=0, n\4, 2^k*binomial(2*n-6*k, 2*k));

Formula

G.f.: (1-x-2*x^4)/((1-x-2*x^4)^2 - 8*x^5).
a(n) = 2*a(n-1) - a(n-2) + 4*a(n-4) + 4*a(n-5) - 4*a(n-8).

A387647 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(2*k,2*n-4*k).

Original entry on oeis.org

1, 0, 1, 2, 1, 12, 5, 30, 61, 64, 281, 314, 857, 1812, 2701, 7606, 11925, 26376, 55393, 96402, 223985, 405276, 835989, 1726158, 3233133, 6901328, 13260073, 26731882, 54453001, 105630628, 217246237, 427776358, 856449221, 1729791512, 3411468145, 6904065986
Offset: 0

Views

Author

Seiichi Manyama, Sep 04 2025

Keywords

Links

Crossrefs

Cf. A108480, A387648.

Programs

  • Magma
    [&+[2^(n-2*k)* Binomial(2*k, 2*n-4*k): k in [0..Floor (n/2)]]: n in [0..40]]; // Vincenzo Librandi, Sep 06 2025
  • Mathematica
    Table[Sum[2^(n-2*k)*Binomial[2*k,2*n-4*k],{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 06 2025 *)
  • PARI
    a(n) = sum(k=0, n\2, 2^(n-2*k)*binomial(2*k, 2*n-4*k));

Formula

G.f.: (1-x^2-2*x^3)/((1-x^2-2*x^3)^2 - 8*x^5).
a(n) = 2*a(n-2) + 4*a(n-3) - a(n-4) + 4*a(n-5) - 4*a(n-6).

A387648 a(n) = Sum_{k=0..floor(n/3)} 2^(n-3*k) * binomial(2*k,2*n-6*k).

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 1, 12, 4, 1, 30, 60, 9, 56, 280, 225, 106, 840, 1681, 852, 2012, 7393, 8102, 6116, 24089, 48288, 39312, 69889, 206354, 268496, 264993, 715868, 1419892, 1498177, 2407662, 5980620, 8659497, 10078152, 21975496, 42559393, 52699770, 81920920, 178653105
Offset: 0

Views

Author

Seiichi Manyama, Sep 04 2025

Keywords

Links

Crossrefs

Cf. A108480, A387647.

Programs

  • PARI
    a(n) = sum(k=0, n\3, 2^(n-3*k)*binomial(2*k, 2*n-6*k));

Formula

G.f.: (1-x^3-2*x^4)/((1-x^3-2*x^4)^2 - 8*x^7).
a(n) = 2*a(n-3) + 4*a(n-4) - a(n-6) + 4*a(n-7) - 4*a(n-8).
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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