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 1-2 of 2 results.

A121576 Riordan array (2-2*x-sqrt(1-8*x+4*x^2), (1-2*x-sqrt(1-8*x+4*x^2))/2).

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 24, 24, 8, 1, 114, 123, 51, 11, 1, 600, 672, 312, 87, 14, 1, 3372, 3858, 1914, 618, 132, 17, 1, 19824, 22992, 11904, 4218, 1068, 186, 20, 1, 120426, 140991, 75183, 28383, 8043, 1689, 249, 23, 1, 749976, 884112, 481704, 190347, 58398, 13929, 2508, 321, 26, 1
Offset: 0

Views

Author

Paul Barry, Aug 08 2006

Keywords

Comments

Inverse of Riordan array (1/(1+2*x), x*(1-x)/(1+2*x)).
Row sums are A047891; first column is A054872. Signed version given by A121575.
Triangle T(n,k), 0 <= k <= n, read by rows, given by [2, 1, 3, 1, 3, 1, 3, 1, 3, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 09 2006

Examples

			Triangle begins
     1;
     2,    1;
     6,    5,    1;
    24,   24,    8,   1;
   114,  123,   51,  11,   1;
   600,  672,  312,  87,  14,  1;
  3372, 3858, 1914, 618, 132, 17, 1;
From _Paul Barry_, Apr 27 2009: (Start)
Production matrix is
  2, 1,
  2, 3, 1,
  2, 3, 3, 1,
  2, 3, 3, 3, 1,
  2, 3, 3, 3, 3, 1,
  2, 3, 3, 3, 3, 3, 1,
  2, 3, 3, 3, 3, 3, 3, 1
In general, the production matrix of the inverse of (1/(1-rx),x(1-x)/(1-rx)) is
  -r, 1,
  -r, 1 - r, 1,
  -r, 1 - r, 1 - r, 1,
  -r, 1 - r, 1 - r, 1 - r, 1,
  -r, 1 - r, 1 - r, 1 - r, 1 - r, 1,
  -r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1,
  -r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1 - r, 1 (End)

Links

Programs

  • Magma
    [[(&+[ 2^j*Binomial(n,j)*Binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1))/2: j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018
  • Mathematica
    Flatten[Table[Sum[Binomial[n,i]Binomial[2n-k-i,n](4-9i+3i^2-6(i-1)n+2n^2)/((n-i+2)(n-i+1))2^i,{i,0,n-k}]/2,{n,0,8},{k,0,n}]]
(* Emanuele Munarini, May 18 2011 *)
  • Maxima
    create_list(sum(binomial(n,i)*binomial(2*n-k-i,n)*(4-9*i+3*i^2-6*(i-1)*n+2*n^2)/((n-i+2)*(n-i+1))*2^i,i,0,n-k)/2,n,0,9,k,0,n); /* Emanuele Munarini, May 18 2011 */
  • PARI
    for(n=0,10, for(k=0,n, print1(sum(j=0, n-k, 2^j*binomial(n,j) *binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1)))/2, ", "))) \\ G. C. Greubel, Nov 02 2018

Formula

T(n,k) = [x^(n-k)](1-2*x-2*x^2)*(1+2*x)^n/(1-x)^(n+1) = (1/2)*Sum_{i=0..n-k} binomial(n,i) * binomial(2*n-k-i,n) * (4 - 9*i + 3*i^2 - 6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i. - Emanuele Munarini, May 18 2011

A121574 Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 16, 8, 1, 16, 44, 37, 11, 1, 32, 112, 134, 67, 14, 1, 64, 272, 424, 305, 106, 17, 1, 128, 640, 1232, 1168, 584, 154, 20, 1, 256, 1472, 3376, 3992, 2641, 998, 211, 23, 1, 512, 3328, 8864, 12592, 10442, 5221, 1574, 277, 26, 1
Offset: 0

Views

Author

Paul Barry, Aug 08 2006

Keywords

Comments

Row sums are A006190(n+1); diagonal sums are A077939.
Inverse is A121575.
A generalized Delannoy number triangle.
Antidiagonal sums are A002478. - Philippe Deléham, Nov 10 2011.
From Peter Bala, Feb 07 2024: (Start)
The following remarks assume the row indexing starts at n = 1.
The sequence of row polynomials R(n,x), beginning R(1,x) = 1, R(2,x) = 2 + x, R(3,x) = 4 + 5*x + x^2 , ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd( R(n,x), R(m,x)) = R(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the polynomial sequence {R(n,x): n >= 1} is a divisibility sequence; that is, if n divides m then R(n,x) divides R(m,x) in Z[x]. (End)

Examples

			Triangle begins
   1;
   2,   1;
   4,   5,   1;
   8,  16,   8,   1;
  16,  44,  37,  11,   1;
  32, 112, 134,  67,  14,  1;
  64, 272, 424, 305, 106, 17, 1;

Links

Crossrefs

Cf. Diagonals: A000012, A016789, A080855, A000079, A053220.

Programs

  • GAP
    T:=Flat(List([0..9],n->List([0..n],k->Sum([0..n-k],j->Binomial(k,j)*Binomial(n-j,k)*2^(n-k-j))))); # Muniru A Asiru, Nov 02 2018
  • Magma
    [[(&+[ Binomial(k, j)*Binomial(n-j, k)*2^(n-k-j): j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018
  • Maple
    T:=(n,k)->add(binomial(k,j)*binomial(n-j,k)*2^(n-k-j),j=0..n-k): seq(seq(T(n,k),k=0..n),n=0..9); # Muniru A Asiru, Nov 02 2018
  • Mathematica
    Table[Sum[Binomial[k, j] Binomial[n-j, k] 2^(n-k-j), {j, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 02 2018 *)
  • PARI
    for(n=0,10, for(k=0,n, print1(sum(j=0, n-k, binomial(k, j)* binomial(n-j, k)*2^(n-k-j)), ", "))) \\ G. C. Greubel, Nov 02 2018

Formula

Number array T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k)*2^(n-k-j).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-1). - Philippe Deléham, Nov 10 2011
Recurrence for row polynomials (with row indexing starting at n = 1): R(n,x) = (x + 2)*R(n-1,x) + x*R(n-2,x) with R(1,x) = 1 and R(2,x) = x + 2. - Peter Bala, Feb 07 2024
Showing 1-2 of 2 results.

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