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 1-4 of 4 results.

A156890 Triangle formed by coefficients of the expansion of p(x, n), where p(x,n) = (1+x-x^2)^(n+1)*Sum_{j >= 0} (j+1)^n*(-x + x^2)^j.

Original entry on oeis.org

1, 1, 1, -1, 1, 1, -4, 5, -2, 1, 1, -11, 22, -23, 14, -3, 1, 1, -26, 92, -158, 145, -82, 32, -4, 1, 1, -57, 359, -906, 1265, -1135, 649, -238, 67, -5, 1, 1, -120, 1311, -4798, 9630, -12132, 10163, -5970, 2406, -620, 135, -6, 1, 1, -247, 4540, -24205, 66769, -113626, 131045, -106889, 62261, -26426, 8033, -1517, 268, -7, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 17 2009

Keywords

Comments

Row sums are equal to 1.

Examples

			Irregular triangle begins as:
  1;
  1;
  1,   -1,    1;
  1,   -4,    5,    -2,    1;
  1,  -11,   22,   -23,   14,     -3,     1;
  1,  -26,   92,  -158,  145,    -82,    32,    -4,    1;
  1,  -57,  359,  -906, 1265,  -1135,   649,  -238,   67,   -5,   1;
  1, -120, 1311, -4798, 9630, -12132, 10163, -5970, 2406, -620, 135, -6, 1;

Links

Crossrefs

Cf. A000295, A156896, A156901, A156918.

Programs

  • Mathematica
    p[x_, n_]:= ((1+x-x^2)^(n+1))*Sum[(j+1)^n*(-x+x^2)^j, {j,0,Infinity}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
  • Sage
    def T(n,k): return ( (1+x-x^2)^(n+1)*sum((j+1)^n*(x^2-x)^j for j in (0..2*n+1)) ).series(x, 2*n+3).list()[k]
[1]+flatten([[T(n,k) for k in (0..2*n-2)] for n in (0..12)]) # G. C. Greubel, Jan 06 2022

Formula

T(n, k) = coefficients of the expansion of p(x, n), where p(x,n) = (1+x-x^2)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-x + x^2)^j.
T(n, 1) = (-1)*A000295(n) for n >= 2. - G. C. Greubel, Jan 06 2022

Extensions

Edited by G. C. Greubel, Jan 06 2022

A156901 Triangle formed by coefficients of the expansion of p(x, n), where p(x,n) = (1 + 2*x - x^2)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-2*x + x^2)^j.

Original entry on oeis.org

1, 1, 1, -2, 1, 1, -8, 8, -4, 1, 1, -22, 55, -52, 23, -6, 1, 1, -52, 290, -472, 394, -188, 50, -8, 1, 1, -114, 1265, -3624, 4838, -3668, 1750, -536, 97, -10, 1, 1, -240, 4884, -24092, 49239, -56448, 40664, -19320, 6231, -1360, 180, -12, 1, 1, -494, 17419, -142124, 441625, -730898, 749723, -515944, 247067, -83122, 19673, -3244, 331, -14, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 17 2009

Keywords

Examples

			Irregular triangle begins as:
  1;
  1;
  1,   -2,    1;
  1,   -8,    8,     -4,     1;
  1,  -22,   55,    -52,    23,     -6,     1;
  1,  -52,  290,   -472,   394,   -188,    50,     -8,    1;
  1, -114, 1265,  -3624,  4838,  -3668,  1750,   -536,   97,   -10,   1;
  1, -240, 4884, -24092, 49239, -56448, 40664, -19320, 6231, -1360, 180, -12, 1;

Links

Crossrefs

Cf. A005803, A156890, A156896, A156918.

Programs

  • Mathematica
    p[x_, n_]= (1+2*x-x^2)^(n+1)*Sum[(k+1)^n*(-2*x+x^2)^k, {k,0,Infinity}];
Table[CoefficientList[p[x, n], x], {n,0,10}]//Flatten
  • Sage
    def T(n, k): return ( (1+2*x-x^2)^(n+1)*sum((j+1)^n*(x^2-2*x)^j for j in (0..2*n+1)) ).series(x, 2*n+2).list()[k]
flatten([1]+[[T(n, k) for k in (0..2*n-2)] for n in (1..12)]) # G. C. Greubel, Jan 07 2022

Formula

T(n, k) = coefficients of the expansion of p(x, n), where p(x,n) = (1 + 2*x - x^2)^(n + 1)*Sum_{j >= 0} (j+1)^n*(-2*x + x^2)^j.
T(n, 1) = (-1)*A005803(n) for n >= 2.

Extensions

Edited by G. C. Greubel, Jan 07 2022

A156918 Triangle formed by coefficients of the expansion of p(x,n) = (1+x-x^2)^(n+1)*Sum_{j >= 0} (2*j+1)^n*(-x + x^2)^j.

Original entry on oeis.org

1, 1, -1, 1, 1, -6, 7, -2, 1, 1, -23, 46, -47, 26, -3, 1, 1, -76, 306, -536, 459, -232, 82, -4, 1, 1, -237, 1919, -5046, 6965, -5995, 3109, -958, 247, -5, 1, 1, -722, 11265, -44634, 91730, -113538, 90417, -49398, 17778, -3630, 737, -6, 1, 1, -2179, 62836, -381037, 1099549, -1878718, 2123525, -1658537, 898985, -346886, 93377, -13109, 2200, -7, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 18 2009

Keywords

Comments

Row sums are one.

Examples

			Irregular triangle begins as:
  1;
  1,   -1,     1;
  1,   -6,     7,     -2,     1;
  1,  -23,    46,    -47,    26,      -3,     1;
  1,  -76,   306,   -536,   459,    -232,    82,     -4,     1;
  1, -237,  1919,  -5046,  6965,   -5995,  3109,   -958,   247,    -5,   1;
  1, -722, 11265, -44634, 91730, -113538, 90417, -49398, 17778, -3630, 737, -6, 1;

Links

Crossrefs

Cf. A060188, A156896, A156890, A156901.

Programs

  • Mathematica
    p[x_, n_] = (1+x-x^2)^(n+1)*Sum[(2*k+1)^n*(-x+x^2)^k, {k, 0, Infinity}];
Table[CoefficientList[p[x, n], x], {n,0,10}]//Flatten
  • Sage
    def T(n, k): return ( (1+x-x^2)^(n+1)*sum((2*j+1)^n*(x^2-x)^j for j in (0..2*n+1)) ).series(x, 2*n+2).list()[k]
flatten([1]+[[T(n, k) for k in (0..2*n)] for n in (1..12)]) # G. C. Greubel, Jan 07 2022

Formula

T(n, k) = coefficients of the expansion of p(x, n), where p(x,n) = (1+x-x^2)^(n + 1)*Sum_{j >= 0} (2*j+1)^n*(-x + x^2)^j.
T(n, 1) = (-1)*A060188(n), for n >= 2. - G. C. Greubel, Jan 07 2022

Extensions

Edited by G. C. Greubel, Jan 07 2022

A156985 Triangle formed by coefficients of the expansion of p(x,n) = (1-x)^(2*n + 1)*Sum_{j >= 0} (1 +j +j^2)^n * x^j.

Original entry on oeis.org

1, 1, 0, 1, 1, 4, 14, 4, 1, 1, 20, 175, 328, 175, 20, 1, 1, 72, 1708, 9784, 17190, 9784, 1708, 72, 1, 1, 232, 14189, 199616, 884498, 1431728, 884498, 199616, 14189, 232, 1, 1, 716, 108250, 3353948, 31986447, 115907544, 176287788, 115907544, 31986447, 3353948, 108250, 716, 1
Offset: 0

Views

Author

Roger L. Bagula, Feb 20 2009

Keywords

Examples

			Irregular triangle begins as:
  1;
  1,   0,     1;
  1,   4,    14,      4,      1;
  1,  20,   175,    328,    175,      20,      1;
  1,  72,  1708,   9784,  17190,    9784,   1708,     72,     1;
  1, 232, 14189, 199616, 884498, 1431728, 884498, 199616, 14189, 232, 1;

Links

Crossrefs

Cf. A010050, A156896, A156890, A156901, A156918.

Programs

  • Mathematica
    p[x_, n_] = (1-x)^(2*n+1)*Sum[(1+k+k^2)^n*x^k, {k, 0, Infinity}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
  • Sage
    def T(n, k): return ( (1-x)^(2*n+1)*sum((j^2+j+1)^n*x^j for j in (0..2*n+1)) ).series(x, 2*n+2).list()[k]
flatten([1]+[[T(n, k) for k in (0..2*n)] for n in (1..12)]) # G. C. Greubel, Jan 07 2022

Formula

T(n, k) = coefficients of the expansion of p(x, n), where p(x,n) = (1-x)^(2*n + 1)*Sum_{j >= 0} (1 +j +j^2)^n * x^j.
Sum_{k=0..2*n} T(n, k) = A010050(n).

Extensions

Edited by G. C. Greubel, Jan 07 2022
Showing 1-4 of 4 results.

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

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