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.

A238236 Expansion of (1-x-x^2)/((x-1)*(x^3+3*x^2+2*x-1)).

Original entry on oeis.org

1, 2, 6, 18, 55, 169, 520, 1601, 4930, 15182, 46754, 143983, 443409, 1365520, 4205249, 12950466, 39882198, 122821042, 378239143, 1164823609, 3587185688, 11047081345, 34020543362, 104769516446, 322647744322, 993624581343, 3059961912097, 9423445312544
Offset: 0

Views

Author

Philippe Deléham, Feb 20 2014

Keywords

Comments

Row sums of the triangle in A152440.

Links

Crossrefs

Cf. A097472, A152440, A099098 (first differences).

Programs

  • Mathematica
    CoefficientList[Series[(1 - x - x^2)/(1 - 3 x - x^2 + 2 x^3 + x^4), {x, 0, 40}], x ](* Vincenzo Librandi, Feb 22 2014 *)

Formula

G.f.: (1-x-x^2)/(1-3*x-x^2+2*x^3+x^4).
a(n) = 3*a(n-1) + a(n-2) -2*a(n-3) - a(n-4), a(0) = 1, a(1) = 2, a(2) = 6, a(3) = 18.
a(n) = A097472(n) - A097472(n-1) - A097472(n-2).
a(n) = A060945(2*n).
a(n)-a(n-1) = A099098(n). - R. J. Mathar, Jun 17 2020

A238241 Riordan array (1/(1-x-x^2)^2, x/(1-x-x^2)^2).

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 10, 14, 6, 1, 20, 40, 27, 8, 1, 38, 105, 98, 44, 10, 1, 71, 256, 315, 192, 65, 12, 1, 130, 594, 924, 726, 330, 90, 14, 1, 235, 1324, 2534, 2472, 1430, 520, 119, 16, 1, 420, 2860, 6588, 7776, 5522, 2535, 770, 152, 18, 1, 744, 6020, 16407, 22968
Offset: 0

Views

Author

Philippe Deléham, Feb 20 2014

Keywords

Comments

Row sums are A097472(n).

Examples

			Triangle begins:
1;
2, 1;
5, 4, 1;
10, 14, 6, 1;
20, 40, 27, 8, 1;
38, 105, 98, 44, 10, 1;
71, 256, 315, 192, 65, 12, 1;
130, 594, 924, 726, 330, 90, 14, 1;
...

Crossrefs

Cf. A037027, A152440
Cf. Diagonals: A000012, A005843, A014106
Cf. Columns: A001629, A001872, A001874

Programs

  • Mathematica
    T[0, 0] = 1; T[n_, k_] := SeriesCoefficient[-1/(x*y - x^4 - 2*x^3 + x^2 + 2*x - 1), {x, 0, n}, {y, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Vladimir Kruchinin *)

Formula

T(n,k) = A037027(n+k+1, 2*k+1).
T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k) - 2*T(n-3,k) - T(n-4,k), T(0,0) = 1, T(n,k) = 0 if k<0 or if k>n.
G.f.: -1/(x*y-x^4-2*x^3+x^2+2*x-1). - Vladimir Kruchinin, Apr 29 2015
Showing 1-2 of 2 results.

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

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