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 1-3 of 3 results.

A099254 Self-convolution of A010892. The g.f. is 1/(Alexander polynomial of granny knot).

Original entry on oeis.org

1, 2, 1, -2, -4, -2, 3, 6, 3, -4, -8, -4, 5, 10, 5, -6, -12, -6, 7, 14, 7, -8, -16, -8, 9, 18, 9, -10, -20, -10, 11, 22, 11, -12, -24, -12, 13, 26, 13, -14, -28, -14, 15, 30, 15, -16, -32, -16, 17, 34, 17, -18, -36, -18, 19, 38, 19, -20, -40, -20, 21, 42, 21
Offset: 0

Views

Author

Paul Barry, Oct 08 2004

Keywords

Comments

A granny knot sequence.
INVERTi transform of A077855: (1, 3, 6, 11, 20, 36, 64, 133, ...). - Gary W. Adamson, Jan 13 2017

Links

Crossrefs

Row sums of array A128502.
Cf. A077855, A076118 (first differences).

Programs

  • Maple
    A099254 := proc(n)
    option remember ;
    if n <= 3 then
        op(n+1,[1,2,1,-2]) ;
    else
        2*procname(n-1)-3*procname(n-2)+2*procname(n-3)-procname(n-4) ;
    end if;
end proc:
seq(A099254(n),n=0..80) ; # R. J. Mathar, Jul 08 2022
  • Mathematica
    LinearRecurrence[{2, -3, 2, -1}, {1, 2, 1, -2}, 100] (* Jean-François Alcover, Sep 21 2022 *)
  • Python
    a0,a1,a2,a3,n = -2,1,2,1,3
print(0,a3)
print(1,a2)
print(2,a1)
print(3,a0)
while n < 20000:
    a0,a1,a2,a3,n = 2*a0-3*a1+2*a2-a3,a0,a1,a2,n+1
    print(n,a0) # A.H.M. Smeets, Sep 13 2018
  • Python
    def A099254(n):
    a, b = divmod(n,3)
    return (1+(b&1))*(-a-1 if a&1 else a+1) # Chai Wah Wu, Jan 31 2023

Formula

G.f.: 1/(1 - 2*x + 3*x^2 - 2*x^3 + x^4) = 1/(1 - x + x^2)^2.
a(n) = 4*sqrt(3)*sin(Pi*n/3 + Pi/3)/9 + 2*(n + 1)*sin(Pi*n/3 + Pi/6)/3.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*(n-k+1)*(-1)^k. - Paul Barry, Nov 12 2004
a(n) = 2*cos(2*Pi*(n + 2)/3)*(floor(n/3) + 1)*(-1)^(n+1). - Tani Akinari, Jul 01 2013
a(n) = (1/54)*(18*(n + 2)*(-1)^floor(n/3) + (3*n + 11)*(-1)^floor((n + 1)/3) - 9*(n + 1)*(-1)^floor((n + 2)/3) - 2*(3*n + 8)*(-1)^floor((n + 4)/3)). - John M. Campbell, Dec 23 2016
From A.H.M. Smeets, Sep 13 2018: (Start)
a(n) = 2*a(n-1) - 3*a(n-2) + 2*a(n-3) - a(n-4) for n >= 4.
a(3*k) = a(3*k+2) = (-1)^k*(k + 1) for k >= 0.
a(3*k+1) = -(-1)^k*2*(k + 1) for k >= 0. (End)
Sum_{n>=0} 1/a(n) = 5*log(2)/2. - Amiram Eldar, May 10 2025

A128503 Array for second (k=2) convolution of Chebyshev's S(n,x)=U(n,x/2) polynomials.

Original entry on oeis.org

1, 3, 6, -3, 10, -12, 15, -30, 6, 21, -60, 30, 28, -105, 90, -10, 36, -168, 210, -60, 45, -252, 420, -210, 15, 55, -360, 756, -560, 105, 66, -495, 1260, -1260, 420, -21, 78, -660, 1980, -2520, 1260, -168, 91, -858, 2970, -4620, 3150, -756, 28, 105, -1092, 4290, -7920, 6930, -2520, 252, 120, -1365
Offset: 0

Views

Author

Wolfdieter Lang Apr 04 2007

Keywords

Comments

S2(n,x):=sum(S(n-k,x)*S1(k,x),k=0..n)= sum(a(n,m)*x^(n-2*m),m=0..floor(n/2)) with the first convolution S1(n,x) given by array A128502.
Row polynomials P2(n,x):= sum(a(n,m)*x^m,m=0..floor(n/2)) (increasing powers of x).

Examples

			n=4: [15,-30,6] stands for the polynomial S2(4,x) = 15*x^4-30*x^2+6 = 2*(S(4,x)*S1(0,x)+S(3,x)*S1(1,x))+S(2,x)*S1(2,x).
n=4:[15,-30,6] stands also for the row polynomial P2(4,x) = 15-30*x+6*x^2.
[1];[3];[6,-3];[10,-12];[15,-30,6];[21,-60,30];[28,-105,90,-10];...

Links

Crossrefs

Row sums (signed array) give A128504. Unsigned row sums are A001628.
Cf. A128502 (k=1 convolution). A128505 (k=3 convolution).

Formula

a(n,m)= binomial(n-m+2,2)*binomial(n-m,m)*(-1)^m, m=0..floor(n/2), n>=0.
a(n,m)= binomial(m+2,2)*binomial(n-m+2,m+2)*(-1)^m, m=0..floor(n/2), n>=0.
G.f. for S2(n,x): 1/(1-x*z+z^2)^3.
G.f. for P2(n,x): 1/(1-z+x*z^2)^3

A294519 Convolution triangle for Chebyshev S polynomials (rising powers).

Original entry on oeis.org

1, 0, 2, -2, 0, 3, 0, -6, 0, 4, 3, 0, -12, 0, 5, 0, 12, 0, -20, 0, 6, -4, 0, 30, 0, -30, 0, 7, 0, -20, 0, 60, 0, -42, 0, 8, 5, 0, -60, 0, 105, 0, -56, 0, 9, 0, 30, 0, -140, 0, 168, 0, -72, 0, 10, -6, 0, 105, 0, -280, 0, 252, 0, -90, 0, 11, 0, -42, 0, 280, 0, -504, 0, 360, 0, -110, 0, 12, 7, 0, -168, 0, 630, 0, -840, 0, 495, 0, -132, 0, 13
Offset: 0

Views

Author

Wolfdieter Lang, Nov 07 2017

Keywords

Comments

See the array A128502 without zeros and falling powers. This is the main entry.
The coefficient triangle for Chebyshev S polynomials is given in A049310.
The self-convolution (or first convolution) of the S polynomials is S1(n, x) := Sum_{k=0..n} S(k, x)*S(n-k, x), n >= 0, and S1(n, x) = Sum_{m=0..n} T(n, m)*x^m.

Examples

			The triangle T(n, m) begins:
n\m   0   1    2    3    4    5    6   7    8    9   10 11 12 ...
0:    1
1:    0   2
2:   -2   0    3
3:    0  -6    0    4
4:    3   0  -12    0    5
5:    0  12    0  -20    0    6
6:   -4   0   30    0  -30    0    7
7:    0 -20    0   60    0  -42    0   8
8:    5   0  -60    0  105    0  -56   0    9
9:    0  30    0 -140    0  168    0 -72    0   10
10:  -6   0  105    0 -280    0  252   0  -90    0   11
11:   0 -42    0  280    0 -504    0 360    0 -110    0 12
12:   7   0 -168    0  630    0 -840   0  495    0 -132  0 13
...

Crossrefs

Cf. A049310, A128502.

Formula

T(n, m) = [x^m] S1(n, x), with the first convolution S1 of the Chebyshev S polynomials. See a comment above.
T(n, m) = 0 if n-m is odd and T(n, m) = (-1)^((n-m)/2)*((n-m)/2 + 1)*binomial(n - (n-m)/2 +1, (n-m)/2 +1) = (-1)^((n-m)/2)*(n - (n-m)/2 + 1)* binomial(n - (n-m)/2, (n-m)/2) if n-m is even.
O.g.f. of {S1(n, x)}_{n >= 0} is G1(z,x) = (1/(1 - x*z + z^2))^2.
Showing 1-3 of 3 results.

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