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 1-5 of 5 results.

A367023 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = hypergeom([1/2, -n - 1, -n], [2, 2], 4*x).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 12, 5, 1, 10, 40, 50, 14, 1, 15, 100, 250, 210, 42, 1, 21, 210, 875, 1470, 882, 132, 1, 28, 392, 2450, 6860, 8232, 3696, 429, 1, 36, 672, 5880, 24696, 49392, 44352, 15444, 1430, 1, 45, 1080, 12600, 74088, 222264, 332640, 231660, 64350, 4862
Offset: 0

Views

Author

Peter Luschny, Nov 06 2023

Keywords

Examples

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,  1;
  [2] 1,  3,    2;
  [3] 1,  6,   12,     5;
  [4] 1, 10,   40,    50,    14;
  [5] 1, 15,  100,   250,   210,     42;
  [6] 1, 21,  210,   875,  1470,    882,    132;
  [7] 1, 28,  392,  2450,  6860,   8232,   3696,    429;
  [8] 1, 36,  672,  5880, 24696,  49392,  44352,  15444,  1430;
  [9] 1, 45, 1080, 12600, 74088, 222264, 332640, 231660, 64350, 4862;

Crossrefs

Cf. A128088 (row sums), A358368 (central terms), A367022.

Programs

  • Maple
    p := n -> hypergeom([1/2, -n - 1, -n], [2, 2], 4*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
  • Mathematica
    T[n_,k_]:=Binomial[n,k]*Binomial[n+1,k]*Binomial[2*k,k]/(k+1)^2;Flatten[Table[T[n,k],{n,0,9},{k,0,n}]]
(* Detlef Meya, Nov 22 2023 *)

Formula

T(2*n, n) = Sum_{k=0..n} CatalanNumber(n)^2 * binomial(n + k, k).
From Detlef Meya, Nov 22 2023: (Start)
T(n, k) = binomial(n, k)*binomial(n+1, k)*binomial(2*k, k)/(k+1)^2.
T(n, k) = A001263(n+1, k+1)*binomial(2*k, k)/(k+1). (End)

A367177 Triangle read by rows, T(n, k) = [x^k] hypergeom([1/2, -n, -n], [1, 1], 4*x).

Original entry on oeis.org

1, 1, 2, 1, 8, 6, 1, 18, 54, 20, 1, 32, 216, 320, 70, 1, 50, 600, 2000, 1750, 252, 1, 72, 1350, 8000, 15750, 9072, 924, 1, 98, 2646, 24500, 85750, 111132, 45276, 3432, 1, 128, 4704, 62720, 343000, 790272, 724416, 219648, 12870
Offset: 0

Views

Author

Peter Luschny, Nov 07 2023

Keywords

Examples

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,   2;
  [2] 1,   8,    6;
  [3] 1,  18,   54,     20;
  [4] 1,  32,  216,    320,      70;
  [5] 1,  50,  600,   2000,    1750,     252;
  [6] 1,  72, 1350,   8000,   15750,    9072,     924;
  [7] 1,  98, 2646,  24500,   85750,  111132,   45276,    3432;
  [8] 1, 128, 4704,  62720,  343000,  790272,  724416,  219648,   12870;
  [9] 1, 162, 7776, 141120, 1111320, 4000752, 6519744, 4447872, 1042470, 48620;

Crossrefs

Cf. A002893 (row sum), A002897 (central column), A000984 (main diagonal).
Cf. A367022, A367023, A387024.

Programs

  • Maple
    p := n -> hypergeom([1/2, -n, -n], [1, 1], 4*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Formula

T(n, k) = binomial(n, k)^2 * binomial(2*k, k).

A367024 Triangle read by rows, T(n, k) = [x^k] -hypergeom([-1/2, -n, -n], [1, 1], 4*x).

Original entry on oeis.org

-1, -1, 2, -1, 8, 2, -1, 18, 18, 4, -1, 32, 72, 64, 10, -1, 50, 200, 400, 250, 28, -1, 72, 450, 1600, 2250, 1008, 84, -1, 98, 882, 4900, 12250, 12348, 4116, 264, -1, 128, 1568, 12544, 49000, 87808, 65856, 16896, 858, -1, 162, 2592, 28224, 158760, 444528, 592704, 342144, 69498, 2860
Offset: 0

Views

Author

Peter Luschny, Nov 07 2023

Keywords

Examples

			Triangle T(n, k) starts:
  [0] -1;
  [1] -1,   2;
  [2] -1,   8,    2;
  [3] -1,  18,   18,     4;
  [4] -1,  32,   72,    64,     10;
  [5] -1,  50,  200,   400,    250,     28;
  [6] -1,  72,  450,  1600,   2250,   1008,     84;
  [7] -1,  98,  882,  4900,  12250,  12348,   4116,    264;
  [8] -1, 128, 1568, 12544,  49000,  87808,  65856,  16896,   858;
  [9] -1, 162, 2592, 28224, 158760, 444528, 592704, 342144, 69498, 2860;

Crossrefs

Cf. A246065 (row sums), -A002420 and A284016 (main diagonal).
Cf. A367022, A367023, A387025.

Programs

  • Maple
    p := n -> -hypergeom([-1/2, -n, -n], [1, 1], 4*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Formula

T(n, k) = binomial(n, k)^2 * binomial(2*k, k) / (2*k - 1).

A367025 Triangle read by rows, T(n, k) = [x^k] p(n), where p(n) = (1 - hypergeom([-1/2, -n - 1, -n - 1], [1, 1], 4*x)) / (2*x).

Original entry on oeis.org

1, 4, 1, 9, 9, 2, 16, 36, 32, 5, 25, 100, 200, 125, 14, 36, 225, 800, 1125, 504, 42, 49, 441, 2450, 6125, 6174, 2058, 132, 64, 784, 6272, 24500, 43904, 32928, 8448, 429, 81, 1296, 14112, 79380, 222264, 296352, 171072, 34749, 1430
Offset: 0

Views

Author

Peter Luschny, Nov 07 2023

Keywords

Examples

			Triangle T(n, k) starts:
  [0]   1;
  [1]   4,    1;
  [2]   9,    9,     2;
  [3]  16,   36,    32,      5;
  [4]  25,  100,   200,    125,     14;
  [5]  36,  225,   800,   1125,    504,      42;
  [6]  49,  441,  2450,   6125,   6174,    2058,     132;
  [7]  64,  784,  6272,  24500,  43904,   32928,    8448,    429;
  [8]  81, 1296, 14112,  79380, 222264,  296352,  171072,  34749,   1430;
  [9] 100, 2025, 28800, 220500, 889056, 1852200, 1900800, 868725, 143000, 4862;

Crossrefs

Cf. A000290 (first column), A000108 (main diagonal).
Cf. A367022, A367023, A387024.

Programs

  • Maple
    p := n -> (1 - hypergeom([-1/2, -n-1, -n-1], [1, 1], 4*x)) / (2*x):
T := (n, k) -> coeff(simplify(p(n)), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
  • Mathematica
    T[n_,k_]:=Binomial[n+1,n-k]^2*Binomial[2*k,k]/(k+1);Flatten[Table[T[n,k],{n,0,9},{k,0,n}]] (* Detlef Meya, Nov 19 2023 *)

Formula

T(n,k) = binomial(n+1,n-k)^2*binomial(2*k,k)/(k+1). - Detlef Meya, Nov 19 2023

A367178 Triangle read by rows. T(n, k) = binomial(n, k)^2 * CatalanNumber(k).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 9, 18, 5, 1, 16, 72, 80, 14, 1, 25, 200, 500, 350, 42, 1, 36, 450, 2000, 3150, 1512, 132, 1, 49, 882, 6125, 17150, 18522, 6468, 429, 1, 64, 1568, 15680, 68600, 131712, 103488, 27456, 1430, 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862
Offset: 0

Views

Author

Peter Luschny, Nov 07 2023

Keywords

Examples

			Triangle T(n, k) starts:
  [0] 1;
  [1] 1,  1;
  [2] 1,  4,    2;
  [3] 1,  9,   18,     5;
  [4] 1, 16,   72,    80,     14;
  [5] 1, 25,  200,   500,    350,     42;
  [6] 1, 36,  450,  2000,   3150,   1512,    132;
  [7] 1, 49,  882,  6125,  17150,  18522,   6468,    429;
  [8] 1, 64, 1568, 15680,  68600, 131712, 103488,  27456,   1430;
  [9] 1, 81, 2592, 35280, 222264, 666792, 931392, 555984, 115830, 4862;

Crossrefs

Cf. A086618 (row sums), A186415 (central column), A000108 (main diagonal).
Cf. A098474, A367022, A367023, A387024, A387024, A367177.

Programs

  • Maple
    T := (n, k) -> binomial(n, k)^2 * binomial(2*k, k) / (k + 1):
seq(seq(T(n, k), k = 0..n), n = 0..9);

Formula

T(n, k) = binomial(n, k)^2 * binomial(2*k, k) / (k + 1).
T(n, k) = [x^n] hypergeom([1/2, -n, -n], [1, 2], 4*x).
Showing 1-5 of 5 results.

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

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