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.

A176021 Triangle T(n, k) = 1 - (-1)^n*(n! + 1) + A176013(n, k) + A176013(n, n-k+1) read by rows.

Original entry on oeis.org

1, 1, 1, 1, -10, 1, 1, 72, 72, 1, 1, -528, -678, -528, 1, 1, 4770, 6780, 6780, 4770, 1, 1, -48025, -87568, -68458, -87568, -48025, 1, 1, 524384, 1287776, 947520, 947520, 1287776, 524384, 1, 1, -6169282, -19982590, -18010942, -10305790, -18010942, -19982590, -6169282, 1
Offset: 1

Views

Author

Roger L. Bagula, Apr 06 2010

Keywords

Comments

Row sums are: 1, 2, -8, 146, -1732, 23102, -339642, 5519362, -98631416, 1926628022, ...

Examples

			Triangle begins as:
  1;
  1,        1;
  1,      -10,         1;
  1,       72,        72,         1;
  1,     -528,      -678,      -528,         1;
  1,     4770,      6780,      6780,      4770,         1;
  1,   -48025,    -87568,    -68458,    -87568,    -48025,         1;
  1,   524384,   1287776,    947520,    947520,   1287776,    524384,        1;
  1, -6169282, -19982590, -18010942, -10305790, -18010942, -19982590, -6169282, 1;

Links

Crossrefs

Cf. A008297, A176013, A176022.

Programs

  • Magma
    A176013:= func< n,k | (-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) >;
[1 - (-1)^n*(Factorial(n) + 1) + A176013(n, k) + A176013(n, n-k+1) : k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 08 2021
  • Mathematica
    A176013[n_, k_]:= (-1)^n*(n!/(k*k!))*Binomial[n-1, k-1]*Binomial[n, k-1];
T[n_, m_]:= 1 - (-1)^n*(n! + 1) + A176013[n, k] + A176013[n, n-k+1];
Table[T[n, k], {n, 12}, {k, n}]//Flatten
  • Sage
    def A176013(n, k): return (-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1)
flatten([[1 - (-1)^n*(factorial(n) + 1) + A176013(n, k) + A176013(n, n-k+1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 08 2021

Formula

T(n, k) = 1 - (-1)^n*(n! + 1) + A176013(n, k) + A176013(n, n-k+1).
T(n, k) = 1 - (-1)^n*(n! + 1) + binomial(n+1, k)*( A008297(n, k) + A008297(n, n-k+1) )/(n+1). - G. C. Greubel, Feb 08 2021

Extensions

Edited by G. C. Greubel, Feb 08 2021

A176022 Triangle T(n, k) = A176013(n, k) + A176013(n, n-k+1), read by rows.

Original entry on oeis.org

-2, 3, 3, -7, -18, -7, 25, 96, 96, 25, -121, -650, -800, -650, -121, 721, 5490, 7500, 7500, 5490, 721, -5041, -53067, -92610, -73500, -92610, -53067, -5041, 40321, 564704, 1328096, 987840, 987840, 1328096, 564704, 40321, -362881, -6532164, -20345472, -18373824, -10668672, -18373824, -20345472, -6532164, -362881
Offset: 1

Views

Author

Roger L. Bagula, Apr 06 2010

Keywords

Examples

			Triangle begins as:
     -2;
      3,      3;
     -7,    -18,      -7;
     25,     96,      96,     25;
   -121,   -650,    -800,   -650,   -121;
    721,   5490,    7500,   7500,   5490,     721;
  -5041, -53067,  -92610, -73500, -92610,  -53067,  -5041;
  40321, 564704, 1328096, 987840, 987840, 1328096, 564704, 40321;

Links

Crossrefs

Cf. A008297, A176013, A176021.

Programs

  • Magma
    A176013:= func< n, k | (-1)^n*(Factorial(n)/(k*Factorial(k)))*Binomial(n-1, k-1)*Binomial(n, k-1) >;
[A176013(n, k) + A176013(n, n-k+1) : k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 15 2021
  • Mathematica
    (* First program *)
T[n_, m_]:= ((-1)^n*n!/(m*m!))*Binomial[n-1, m-1]*Binomial[n, m-1] + ((-1)^n*n!)/((n-m+1)*(n-m+1)!)*Binomial[n-1, n-m] Binomial[n, n-m];
Table[T[n, m], {n,10}, {m,n}]//Flatten
(* Second program *)
A176013[n_, k_] := (-1)^n*(n!/(k*k!))*Binomial[n-1, k-1]*Binomial[n, k-1];
T[n_, k_]:= A176013[n, k] + A176013[n, n-k+1];
Table[T[n, k], {n, 10}, {k, n}]//Flatten (* G. C. Greubel, Feb 15 2021 *)
  • Sage
    def A176013(n, k): return (-1)^n*(factorial(n)/(k*factorial(k)))*binomial(n-1, k-1)*binomial(n, k-1)
flatten([[A176013(n, k) + A176013(n, n-k+1) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 15 2021

Formula

T(n, k) = ((-1)^n*n!/(k*k!))*binomial(n-1, k-1)*binomial(n, k-1) + ((-1)^n*n!)/((n-k+1)*(n-k+1)!)*binomial(n-1, n-k)*binomial(n, n-k).
From G. C. Greubel, Feb 15 2021: (Start)
T(n, k) = A176013(n, k) + A176013(n, n-k+1), where A176013(n, k) = (-1)^n*(n!/(k*k!))*binomial(n-1, k-1)*binomial(n, k-1).
Sum_{k=1..n} T(n, k) = 2*(-1)^n * n! * Hypergeometric2F2(-n, -(n-1); 2, 2; 1). (End)

Extensions

Edited by G. C. Greubel, Feb 15 2021

A174696 Triangle T(n, k) = n!*(1/k)^2*(binomial(n-1, k-1)*binomial(n, k-1))^2 - n! + 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 49, 1, 1, 841, 841, 1, 1, 11881, 47881, 11881, 1, 1, 161281, 1799281, 1799281, 161281, 1, 1, 2217601, 55560961, 154344961, 55560961, 2217601, 1, 1, 31570561, 1548892801, 9680791681, 9680791681, 1548892801, 31570561, 1, 1, 469929601, 40967337601, 501853968001, 1129171881601, 501853968001, 40967337601, 469929601, 1
Offset: 1

Views

Author

Roger L. Bagula, Mar 27 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,        1;
  1,       49,          1;
  1,      841,        841,          1;
  1,    11881,      47881,      11881,          1;
  1,   161281,    1799281,    1799281,     161281,          1;
  1,  2217601,   55560961,  154344961,   55560961,    2217601,        1;
  1, 31570561, 1548892801, 9680791681, 9680791681, 1548892801, 31570561, 1;

Links

Crossrefs

Cf. A174158, A174694, A176013, A319743.

Programs

  • Magma
    A174696:= func< n, k | (Factorial(n)/k^2)*(Binomial(n-1, k-1)*Binomial(n, k-1))^2 - Factorial(n) + 1 >;
[A174696(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 09 2021
  • Mathematica
    T[n_, m_]:= n!*(1/k)^2*(Binomial[n-1, k-1]*Binomial[n, k-1])^2 - n! + 1;
Table[T[n, k], {n,12}, {k,n}]//Flatten
  • Sage
    def A174696(n, k): return (factorial(n)/k^2)*(binomial(n-1, k-1)*binomial(n, k-1))^2 - factorial(n) + 1
flatten([[A174696(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 09 2021

Formula

T(n, k) = n!*(1/k)^2*(binomial(n-1, k-1)*binomial(n, k-1))^2 - n! + 1.
From G. C. Greubel, Feb 09 2021: (Start)
T(n, k) = n! * A174158(n, k) - n! + 1.
Sum_{k=1..n} T(n,k) = n! * Hypergeometric4F3([-n, -n, 1-n, 1-n], [1, 2, 2], 1) - n*(n! - 1) = n! * A319743(n) - n*(n! - 1). (End)

Extensions

Edited by G. C. Greubel, Feb 09 2021

A174694 Triangle T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 13, 1, 1, 121, 121, 1, 1, 1081, 2281, 1081, 1, 1, 10081, 35281, 35281, 10081, 1, 1, 100801, 524161, 876961, 524161, 100801, 1, 1, 1088641, 7862401, 19716481, 19716481, 7862401, 1088641, 1, 1, 12700801, 121564801, 426384001, 639757441, 426384001, 121564801, 12700801, 1
Offset: 1

Views

Author

Roger L. Bagula, Mar 27 2010

Keywords

Examples

			Triangle begin as:
  1;
  1,        1;
  1,       13,         1;
  1,      121,       121,         1;
  1,     1081,      2281,      1081,         1;
  1,    10081,     35281,     35281,     10081,         1;
  1,   100801,    524161,    876961,    524161,    100801,         1;
  1,  1088641,   7862401,  19716481,  19716481,   7862401,   1088641,        1;
  1, 12700801, 121564801, 426384001, 639757441, 426384001, 121564801, 12700801, 1;

Links

Crossrefs

Cf. A000108, A174696, A176013.

Programs

  • Magma
    A174694:= func< n, k | (Factorial(n)/k)*Binomial(n-1, k-1)*Binomial(n, k-1) - Factorial(n) + 1 >;
[A174694(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Feb 09 2021
  • Mathematica
    T[n_, k_]:= n!*(1/k)*Binomial[n-1, k-1]*Binomial[n, k-1] - n! + 1;
Table[T[n, k], {n,12}, {k,n}]//Flatten
  • Sage
    def A174694(n, k): return (factorial(n)/k)*binomial(n-1, k-1)*binomial(n, k-1) - factorial(n) + 1
flatten([[A174694(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Feb 09 2021

Formula

T(n, k) = n!*(1/k)*binomial(n-1, k-1)*binomial(n, k-1) - n! + 1.
From G. C. Greubel, Feb 09 2021: (Start)
T(n, k) = (-1)^n * k! * A176013(n, k) - n! + 1.
Sum_{k=1..n} T(n,k) = n! * (C_{n} - n) + n, where C_{n} are the Catalan numbers (A000108). (End)

Extensions

Edited by G. C. Greubel, Feb 09 2021
Showing 1-4 of 4 results.

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