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 1-7 of 7 results.

A306914 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k+x^k).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 6, 0, 0, 1, 5, 10, 9, -4, 0, 1, 6, 15, 20, 9, -8, 0, 1, 7, 21, 35, 34, 0, -8, 0, 1, 8, 28, 56, 70, 48, -27, 0, 0, 1, 9, 36, 84, 126, 125, 48, -81, 16, 0, 1, 10, 45, 120, 210, 252, 200, 0, -162, 32, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2019

Keywords

Examples

			Square array begins:
   1,  1,    1,    1,   1,    1,    1,    1, ...
   0,  2,    3,    4,   5,    6,    7,    8, ...
   0,  2,    6,   10,  15,   21,   28,   36, ...
   0,  0,    9,   20,  35,   56,   84,  120, ...
   0, -4,    9,   34,  70,  126,  210,  330, ...
   0, -8,    0,   48, 125,  252,  462,  792, ...
   0, -8,  -27,   48, 200,  461,  924, 1716, ...
   0,  0,  -81,    0, 275,  780, 1715, 3432, ...
   0, 16, -162, -164, 275, 1209, 2989, 6434, ...

Links

Crossrefs

Columns 1-9 give A000007, A099087, A057083, A099589(n+3), A289389(n+4), A306940, (-1)^n * A049018(n), A306941, A306942.
Cf. A039912, A306913, A306915, A307039, A307079, A307394.

Programs

  • Mathematica
    A[n_, k_] := SeriesCoefficient[1/((1-x)^k + x^k), {x, 0, n}];
Table[A[n-k+1, k], {n, 0, 11}, {k, n+1, 1, -1}] // Flatten (* Jean-François Alcover, Mar 20 2019 *)

Formula

A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+k-1,k*j+k-1).
A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - Seiichi Manyama, Apr 07 2019

A306939 Expansion of 1/((1 - x)^9 - x^9).

Original entry on oeis.org

1, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24311, 43776, 75753, 127110, 209475, 346104, 591261, 1081575, 2163150, 4686826, 10656387, 24582663, 56191734, 125640180, 273241161, 577147212, 1184959314, 2369918628, 4631710931, 8881943832, 16798969548, 31537530456
Offset: 0

Views

Author

Seiichi Manyama, Mar 17 2019

Keywords

Links

Crossrefs

Column 9 of A306915.
Cf. A306860.

Programs

  • Mathematica
    CoefficientList[Series[1/((1 - x)^9 - x^9), {x, 0, 30}], x] (* Amiram Eldar, May 25 2021 *)
  • PARI
    {a(n) = sum(k=0, n\9, binomial(n+8, 9*k+8))}
  • PARI
    N=66; x='x+O('x^N); Vec(1/((1-x)^9-x^9))

Formula

a(n) = Sum_{k=0..floor(n/9)} binomial(n+8,9*k+8).
a(n) = 9*a(n-1) - 36*a(n-2) + 84*a(n-3) - 126*a(n-4) + 126*a(n-5) - 84*a(n-6) + 36*a(n-7) - 9*a(n-8) + 2*a(n-9) for n > 8.

A307047 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1+x)^k-x^k).

Original entry on oeis.org

1, 1, 0, 1, -2, 0, 1, -3, 4, 0, 1, -4, 6, -8, 0, 1, -5, 10, -9, 16, 0, 1, -6, 15, -20, 9, -32, 0, 1, -7, 21, -35, 36, 0, 64, 0, 1, -8, 28, -56, 70, -64, -27, -128, 0, 1, -9, 36, -84, 126, -125, 120, 81, 256, 0, 1, -10, 45, -120, 210, -252, 200, -240, -162, -512, 0
Offset: 0

Views

Author

Seiichi Manyama, Mar 21 2019

Keywords

Examples

			Square array begins:
   1,    1,    1,    1,    1,    1,     1,     1, ...
   0,   -2,   -3,   -4,   -5,   -6,    -7,    -8, ...
   0,    4,    6,   10,   15,   21,    28,    36, ...
   0,   -8,   -9,  -20,  -35,  -56,   -84,  -120, ...
   0,   16,    9,   36,   70,  126,   210,   330, ...
   0,  -32,    0,  -64, -125, -252,  -462,  -792, ...
   0,   64,  -27,  120,  200,  463,   924,  1716, ...
   0, -128,   81, -240, -275, -804, -1715, -3432, ...
   0,  256, -162,  496,  275, 1365,  2989,  6436, ...

Links

Crossrefs

Columns 1-7 give A000007, A122803, A000748, (-1)^n * A000749(n+3), A000750, A006090, A049018.
Cf. A039912 (square array A(n,k), n >= 0, k >= 2), A306913, A306914, A306915.

Programs

  • Mathematica
    T[n_, k_] := (-1)^n * Sum[(-1)^(j * Mod[k, 2]) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

Formula

A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^((k mod 2) * j) * binomial(n+k-1,k*j+k-1).

A306913 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1+x)^k+x^k).

Original entry on oeis.org

1, 1, -2, 1, -2, 4, 1, -3, 2, -8, 1, -4, 6, 0, 16, 1, -5, 10, -11, -4, -32, 1, -6, 15, -20, 21, 8, 64, 1, -7, 21, -35, 34, -42, -8, -128, 1, -8, 28, -56, 70, -48, 85, 0, 256, 1, -9, 36, -84, 126, -127, 48, -171, 16, -512, 1, -10, 45, -120, 210, -252, 220, 0, 342, -32, 1024
Offset: 0

Views

Author

Seiichi Manyama, Mar 16 2019

Keywords

Examples

			Square array begins:
      1,  1,    1,    1,    1,    1,     1,     1, ...
     -2, -2,   -3,   -4,   -5,   -6,    -7,    -8, ...
      4,  2,    6,   10,   15,   21,    28,    36, ...
     -8,  0,  -11,  -20,  -35,  -56,   -84,  -120, ...
     16, -4,   21,   34,   70,  126,   210,   330, ...
    -32,  8,  -42,  -48, -127, -252,  -462,  -792, ...
     64, -8,   85,   48,  220,  461,   924,  1716, ...
   -128,  0, -171,    0, -385, -780, -1717, -3432, ...
    256, 16,  342, -164,  715, 1209,  3017,  6434, ...

Links

Crossrefs

Columns 1-2 give A122803, A108520.
Cf. A039912, A306914, A306915.

Programs

  • Mathematica
    A[n_, k_] := (-1)^n * Sum[(-1)^(Mod[k+1, 2] * j) * Binomial[n + k - 1, k*j + k - 1], {j, 0, Floor[n/k]}]; Table[A[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 25 2021 *)

Formula

A(n,k) = (-1)^n * Sum_{j=0..floor(n/k)} (-1)^(((k+1) mod 2) * j) * binomial(n+k-1,k*j+k-1).

A307078 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k-x^k).

Original entry on oeis.org

1, 1, 3, 1, 2, 7, 1, 2, 4, 15, 1, 2, 3, 8, 31, 1, 2, 3, 5, 16, 63, 1, 2, 3, 4, 10, 32, 127, 1, 2, 3, 4, 6, 21, 64, 255, 1, 2, 3, 4, 5, 12, 43, 128, 511, 1, 2, 3, 4, 5, 7, 28, 86, 256, 1023, 1, 2, 3, 4, 5, 6, 14, 64, 171, 512, 2047, 1, 2, 3, 4, 5, 6, 8, 36, 136, 341, 1024, 4095
Offset: 0

Views

Author

Seiichi Manyama, Mar 22 2019

Keywords

Examples

			Square array begins:
     1,   1,   1,   1,  1,  1,  1,  1, 1, ...
     3,   2,   2,   2,  2,  2,  2,  2, 2, ...
     7,   4,   3,   3,  3,  3,  3,  3, 3, ...
    15,   8,   5,   4,  4,  4,  4,  4, 4, ...
    31,  16,  10,   6,  5,  5,  5,  5, 5, ...
    63,  32,  21,  12,  7,  6,  6,  6, 6, ...
   127,  64,  43,  28, 14,  8,  7,  7, 7, ...
   255, 128,  86,  64, 36, 16,  9,  8, 8, ...
   511, 256, 171, 136, 93, 45, 18, 10, 9, ...

Links

Crossrefs

Columns 1-6 give A126646, A000079, A024494(n+1), A038504(n+1), A133476(n+1), A119336.
Cf. A306846, A306915, A307079.

Programs

  • Mathematica
    T[n_, k_] := Sum[Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

Formula

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+1,k*j+1).
A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i,k*j) * binomial(n-i,k*j).

A307393 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k-x^k).

Original entry on oeis.org

1, 1, 5, 1, 4, 16, 1, 4, 11, 42, 1, 4, 10, 26, 99, 1, 4, 10, 21, 57, 219, 1, 4, 10, 20, 42, 120, 466, 1, 4, 10, 20, 36, 84, 247, 968, 1, 4, 10, 20, 35, 64, 169, 502, 1981, 1, 4, 10, 20, 35, 57, 120, 340, 1013, 4017, 1, 4, 10, 20, 35, 56, 93, 240, 682, 2036, 8100
Offset: 0

Views

Author

Seiichi Manyama, Apr 07 2019

Keywords

Examples

			Square array begins:
     1,   1,   1,   1,   1,   1,   1,   1, ...
     5,   4,   4,   4,   4,   4,   4,   4, ...
    16,  11,  10,  10,  10,  10,  10,  10, ...
    42,  26,  21,  20,  20,  20,  20,  20, ...
    99,  57,  42,  36,  35,  35,  35,  35, ...
   219, 120,  84,  64,  57,  56,  56,  56, ...
   466, 247, 169, 120,  93,  85,  84,  84, ...
   968, 502, 340, 240, 165, 130, 121, 120, ...

Links

Crossrefs

Columns 1-5 give A002662(n+3), A125128(n+1), A111927(n+3), A000749(n+3), A139748(n+3).
Cf. A306915, A306846, A307078, A307394.

Programs

  • Mathematica
    T[n_, k_] := Sum[Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

Formula

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+3,k*j+3).
A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i+1,k*j+1) * binomial(n-i+1,k*j+1).

A307665 A(n,k) = Sum_{j=0..floor(n/k)} binomial(2*n,k*j+n), square array A(n,k) read by antidiagonals, for n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 3, 1, 2, 11, 1, 2, 7, 42, 1, 2, 6, 26, 163, 1, 2, 6, 21, 99, 638, 1, 2, 6, 20, 78, 382, 2510, 1, 2, 6, 20, 71, 297, 1486, 9908, 1, 2, 6, 20, 70, 262, 1145, 5812, 39203, 1, 2, 6, 20, 70, 253, 990, 4447, 22819, 155382, 1, 2, 6, 20, 70, 252, 936, 3796, 17358, 89846, 616666
Offset: 0

Views

Author

Seiichi Manyama, Apr 20 2019

Keywords

Examples

			Square array begins:
      1,    1,    1,    1,    1,    1,    1,    1, ...
      3,    2,    2,    2,    2,    2,    2,    2, ...
     11,    7,    6,    6,    6,    6,    6,    6, ...
     42,   26,   21,   20,   20,   20,   20,   20, ...
    163,   99,   78,   71,   70,   70,   70,   70, ...
    638,  382,  297,  262,  253,  252,  252,  252, ...
   2510, 1486, 1145,  990,  936,  925,  924,  924, ...
   9908, 5812, 4447, 3796, 3523, 3446, 3433, 3432, ...

Crossrefs

Columns 1-2 give A032443, A114121.
Cf. A306846, A306915, A307393, A307668.

Programs

  • Mathematica
    T[n_, k_] := Sum[Binomial[2*n, k*j + n], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 13 2021*)
Showing 1-7 of 7 results.

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