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.

A342120 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - k*x - k*x^2).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 3, 0, 1, 4, 12, 16, 5, 0, 1, 5, 20, 45, 44, 8, 0, 1, 6, 30, 96, 171, 120, 13, 0, 1, 7, 42, 175, 464, 648, 328, 21, 0, 1, 8, 56, 288, 1025, 2240, 2457, 896, 34, 0, 1, 9, 72, 441, 1980, 6000, 10816, 9315, 2448, 55, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 28 2021

Keywords

Examples

			Square array begins:
  1, 1,   1,   1,    1,    1, ...
  0, 1,   2,   3,    4,    5, ...
  0, 2,   6,  12,   20,   30, ...
  0, 3,  16,  45,   96,  175, ...
  0, 5,  44, 171,  464, 1025, ...
  0, 8, 120, 648, 2240, 6000, ...

Links

Crossrefs

Columns 0..10 give A000007, A000045(n+1), A002605(n+1), A030195(n+1), A057087, A057088, A057089, A057090, A057091, A057092, A057093.
Rows 0..2 give A000012, A001477, A002378.
Main diagonal gives A109516(n+1).
Cf. A342129, A342133, A342134.

Programs

  • Maple
    T:= (n, k)-> (<<0|1>, >^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 01 2021
  • Mathematica
    T[n_, k_] := Sum[If[k == j == 0, 1, k^j] * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)
  • PARI
    T(n, k) = sum(j=0, n\2, k^(n-j)*binomial(n-j, j));
  • PARI
    T(n, k) = sum(j=0, n, k^j*binomial(j, n-j));
  • PARI
    T(n, k) = round((-sqrt(k)*I)^n*polchebyshev(n, 2, sqrt(k)*I/2));

Formula

T(0,k) = 1, T(1,k) = k and T(n,k) = k*(T(n-1,k) + T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} k^(n-j) * binomial(n-j,j) = Sum_{j=0..n} k^j * binomial(j,n-j).
T(n,k) = (-sqrt(k)*i)^n * S(n, sqrt(k)*i) with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind.

A342133 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2*k*x + k*x^2).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 14, 4, 0, 1, 8, 33, 48, 5, 0, 1, 10, 60, 180, 164, 6, 0, 1, 12, 95, 448, 981, 560, 7, 0, 1, 14, 138, 900, 3344, 5346, 1912, 8, 0, 1, 16, 189, 1584, 8525, 24960, 29133, 6528, 9, 0, 1, 18, 248, 2548, 18180, 80750, 186304, 158760, 22288, 10, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2021

Keywords

Examples

			Square array begins:
  1, 1,   1,    1,     1,     1, ...
  0, 2,   4,    6,     8,    10, ...
  0, 3,  14,   33,    60,    95, ...
  0, 4,  48,  180,   448,   900, ...
  0, 5, 164,  981,  3344,  8525, ...
  0, 6, 560, 5346, 24960, 80750, ...

Links

Crossrefs

Columns 0..5 give A000007, A000027(n+1), A007070, A138395, A099156(n+1), A190987(n+1).
Rows 0..2 give A000012, A005843, A033991.
Main diagonal gives (-1) * A109520(n+1).
Cf. A342120, A342129, A342134.

Programs

  • Maple
    T:= (n, k)-> (<<0|1>, <-k|2*k>>^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 01 2021
  • Mathematica
    T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * (-2)^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 27 2021 *)
  • PARI
    T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*(-2)^(-j)*binomial(n-j, j));
  • PARI
    T(n, k) = sum(j=0, n, (2*k)^j*(-2)^(j-n)*binomial(j, n-j));
  • PARI
    T(n, k) = round(sqrt(k)^n*polchebyshev(n, 2, sqrt(k)));

Formula

T(0,k) = 1, T(1,k) = 2*k and T(n,k) = k*(2*T(n-1,k) - T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (-1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (-1/2)^(n-j) * binomial(j,n-j).
T(n,k) = sqrt(k)^n * U(n, sqrt(k)) where U(n, x) is a Chebyshev polynomial of the second kind.

A342134 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of g.f. 1/(1 - 2*k*x - k*x^2).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 6, 18, 12, 0, 1, 8, 39, 80, 29, 0, 1, 10, 68, 252, 356, 70, 0, 1, 12, 105, 576, 1629, 1584, 169, 0, 1, 14, 150, 1100, 4880, 10530, 7048, 408, 0, 1, 16, 203, 1872, 11525, 41344, 68067, 31360, 985, 0, 1, 18, 264, 2940, 23364, 120750, 350272, 439992, 139536, 2378, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2021

Keywords

Examples

			Square array begins:
  1,  1,    1,     1,     1,      1, ...
  0,  2,    4,     6,     8,     10, ...
  0,  5,   18,    39,    68,    105, ...
  0, 12,   80,   252,   576,   1100, ...
  0, 29,  356,  1629,  4880,  11525, ...
  0, 70, 1584, 10530, 41344, 120750, ...

Links

Crossrefs

Columns 0..5 give A000007, A000129(n+1), A090017(n+1), A090018, A190510(n+1), A190955(n+1).
Rows 0..2 give A000012, A005843, A007742.
Main diagonal gives A109517(n+1).
Cf. A342120, A342129, A342133.

Programs

  • Maple
    T:= (n, k)-> (<<0|1>, >^(n+1))[1, 2]:
seq(seq(T(n, d-n), n=0..d), d=0..12); # Alois P. Heinz, Mar 01 2021
  • Mathematica
    T[n_, k_] := Sum[If[k == j == 0, 1, (2*k)^j] * 2^(j - n) * Binomial[j, n - j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Apr 27 2021 *)
  • PARI
    T(n, k) = sum(j=0, n\2, (2*k)^(n-j)*2^(-j)*binomial(n-j, j));
  • PARI
    T(n, k) = sum(j=0, n, (2*k)^j*2^(j-n)*binomial(j, n-j));
  • PARI
    T(n, k) = round((-sqrt(k)*I)^n*polchebyshev(n, 2, sqrt(k)*I));

Formula

T(0,k) = 1, T(1,k) = 2*k and T(n,k) = k*(2*T(n-1,k) + T(n-2,k)) for n > 1.
T(n,k) = Sum_{j=0..floor(n/2)} (2*k)^(n-j) * (1/2)^j * binomial(n-j,j) = Sum_{j=0..n} (2*k)^j * (1/2)^(n-j) * binomial(j,n-j).
T(n,k) = (-sqrt(k)*i)^n * U(n, sqrt(k)*i) where U(n, x) is a Chebyshev polynomial of the second kind.
Showing 1-3 of 3 results.

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