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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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A307158 a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n,3*k)^2.

Original entry on oeis.org

1, 1, 1, 0, -15, -99, -398, -1175, -2351, 0, 29601, 183195, 756978, 2351805, 4885791, 0, -63746991, -400000275, -1675991918, -5274560891, -11081420615, 0, 147257373891, 931226954949, 3929550225586, 12446852889901, 26304183607651, 0, -353181028924809
Offset: 0

Views

Author

Seiichi Manyama, Mar 27 2019

Keywords

Links

Crossrefs

Central coefficients of number triangle A307156.
Cf. A057681, A119363, A307091.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^k * Binomial[n,3*k]^2, {k, 0, Floor[n/3]}]; Array[a, 30, 0] (* Amiram Eldar, May 20 2021 *)
  • PARI
    {a(n) = sum(k=0, n\3, (-1)^k*binomial(n, 3*k)^2)}

Formula

a(6*n+3) = 0 for n >= 0.

A307395 Expansion of 1/((1 - x) * ((1 - x)^3 + x^3)).

Original entry on oeis.org

1, 4, 10, 19, 28, 28, 1, -80, -242, -485, -728, -728, 1, 2188, 6562, 13123, 19684, 19684, 1, -59048, -177146, -354293, -531440, -531440, 1, 1594324, 4782970, 9565939, 14348908, 14348908, 1, -43046720, -129140162, -258280325, -387420488, -387420488, 1, 1162261468
Offset: 0

Views

Author

Seiichi Manyama, Apr 07 2019

Keywords

Links

Crossrefs

Column 5 of A307394.
Partial sums of A057083.

Programs

  • Mathematica
    LinearRecurrence[{4, -6, 3}, {1, 4, 10}, 38] (* Amiram Eldar, May 13 2021 *)
  • PARI
    {a(n) = sum(k=0, n\3, (-1)^k*binomial(n+3, 3*k+3))}
  • PARI
    N=66; x='x+O('x^N); Vec(1/((1-x)*((1-x)^3+x^3)))

Formula

a(n) = Sum_{k=0..floor(n/3)} (-1)^k*binomial(n+3,3*k+3).
a(n) = 4*a(n-1) - 6*a(n-2) + 3*a(n-3) for n > 2.
a(6*n) = 1.
a(n) = 1 - A057681(n+3). - Yomna Bakr and Greg Dresden, Apr 22 2024

A227430 Expansion of x^2*(1-x)^3/((1-2*x)*(1-x+x^2)*(1-3*x+3x^2)).

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 29, 45, 90, 220, 561, 1365, 3095, 6555, 13110, 25126, 46971, 87381, 164921, 320001, 640002, 1309528, 2707629, 5592405, 11450531, 23166783, 46333566, 91869970, 181348455, 357913941, 708653429, 1410132405, 2820264810, 5662052980
Offset: 0

Views

Author

Paul Curtz, Jul 11 2013

Keywords

Comments

Consider the binomial transform of 0, 0, 0, 0, 0, 1 (period 6) with its differences:
0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n): after 0, it is A192080.
0, 0, 0, 0, 1, 5, 15, 35, 70, 126,... e(n)
0, 0, 0, 1, 4, 10, 20, 35, 56, 85,... f(n)
0, 0, 1, 3, 6, 10, 15, 21, 29, 45,... a(n)
0, 1, 2, 3, 4, 5, 6, 8, 16, 45,... b(n)
1, 1, 1, 1, 1, 1, 2, 8, 29, 85,... c(n)
0, 0, 0, 0, 0, 1, 6, 21, 56, 126,... d(n).
a(n) + d(n) = A024495(n),
b(n) + e(n) = A131708(n),
c(n) + f(n) = A024493(n).
a(n) - d(n) = 0, 0, 1, 3, 6, 9, 9, 0,... A057083(n-2)
b(n) - e(n) = 0, 1, 2, 3, 3, 0, -9, -27,... A057682(n)
c(n) - f(n) = 1, 1, 1, 0, -3, -9, -18, -27,... A057681(n)
d(n) - a(n) = 0, 0, -1, -3, -6, -9, -9, 0,... -A057083(n-2)
e(n) - b(n) = 0, -1, -2, -3, -3, 0, 9, 27,... -A057682(n)
f(n) - c(n) = -1, -1, -1, 0, 3, 9, 18, 27,... -A057681(n).
The first column is A131531(n).
The first two trisections are multiples of 3. Is the third (1, 10, 29,...) mod 9 A029898(n)?

Examples

			a(6)=6*10-15*6+20*3-15*1+6*0=15, a(7)=90-150+120-45+6=21.

Links

Programs

  • Mathematica
    Join[{0},LinearRecurrence[{6,-15,20,-15,6},{0,1,3,6,10},40]] (* Harvey P. Dale, Dec 17 2014 *)
  • PARI
    {a(n) = sum(k=0, n\6, binomial(n, 6*k+2))} \\ Seiichi Manyama, Mar 23 2019

Formula

a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) for n>5, a(0)=a(1)=0, a(2)=1, a(3)=3, a(4)=6, a(5)=10.
a(n) = A024495(n) - A192080(n-5) for n>4.
G.f.: -(x^5 - 3*x^4 + 3*x^3 - x^2)/((1-2*x)*(1-x+x^2)*(1-3*x+3*x^2)). - Ralf Stephan, Jul 13 2013
a(n) = Sum_{k=0..floor(n/6)} binomial(n,6*k+2). - Seiichi Manyama, Mar 23 2019

Extensions

Definition uses the g.f. of Ralf Stephan.
More terms from Harvey P. Dale, Dec 17 2014
Previous Showing 11-13 of 13 results.

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