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.

A326659 T(n,k) = [0=0]; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 15, 18, 6, 1, 64, 132, 96, 24, 1, 325, 980, 1140, 600, 120, 1, 1956, 7830, 12720, 10440, 4320, 720, 1, 13699, 68502, 143850, 162120, 103320, 35280, 5040, 1, 109600, 657608, 1698816, 2447760, 2123520, 1108800, 322560, 40320
Offset: 0

Views

Author

Alois P. Heinz, Sep 12 2019

Keywords

Comments

[] is an Iverson bracket.

Examples

			Triangle T(n,k) begins:
  1;
  1,     1;
  1,     4,     2;
  1,    15,    18,      6;
  1,    64,   132,     96,     24;
  1,   325,   980,   1140,    600,    120;
  1,  1956,  7830,  12720,  10440,   4320,   720;
  1, 13699, 68502, 143850, 162120, 103320, 35280, 5040;
  ...

Links

Crossrefs

Columns k=0-2 give: A000012, A007526, 2*A134432(n-1).
Main diagonal gives A000142.
Row sums give A308876.
Cf. A000522, A073474, A093964, A143409, A196347, A271705, A327606.

Programs

  • Maple
    T:= proc(n, k) option remember;
      `if`(0=0, 1, 0)
    end:
seq(seq(T(n, k), k=0..n), n=0..10);
  • Mathematica
    T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = Boole[0 < k <= n]*n*(T[n-1, k-1] + T[n-1, k]) + Boole[k == 0 && n >= 0];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 09 2021 *)

Formula

E.g.f. of column k: exp(x)*(x/(1-x))^k.
T(n,k) = k! * A271705(n,k).
T(n,k) = n * A073474(n-1,k-1) for n,k >= 1.
T(n,1) = n * A000522(n-1) for n >= 1.
T(n,2) = n * A093964(n-1) for n >= 1.
Sum_{k=1..n} k * T(n,k) = A327606(n).

A174690 Triangle T(n, k) = n!*binomial(n, k) - n! + 1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 13, 13, 1, 1, 73, 121, 73, 1, 1, 481, 1081, 1081, 481, 1, 1, 3601, 10081, 13681, 10081, 3601, 1, 1, 30241, 100801, 171361, 171361, 100801, 30241, 1, 1, 282241, 1088641, 2217601, 2782081, 2217601, 1088641, 282241, 1, 1, 2903041, 12700801, 30119041, 45360001, 45360001, 30119041, 12700801, 2903041, 1
Offset: 0

Views

Author

Roger L. Bagula, Mar 27 2010

Keywords

Examples

			Triangle begins as:
  1;
  1,      1;
  1,      3,       1;
  1,     13,      13,       1;
  1,     73,     121,      73,       1;
  1,    481,    1081,    1081,     481,       1;
  1,   3601,   10081,   13681,   10081,    3601,       1;
  1,  30241,  100801,  171361,  171361,  100801,   30241,      1;
  1, 282241, 1088641, 2217601, 2782081, 2217601, 1088641, 282241, 1;

Links

Crossrefs

Cf. A000165, A021012, A196347.

Programs

  • Magma
    [Factorial(n)*(Binomial(n,k) -1) + 1: k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 09 2021
  • Mathematica
    T[n_, k_]:= n!*Binomial[n, k] - n! + 1;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
  • Sage
    flatten([[factorial(n)*(binomial(n,k) -1) + 1 for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 09 2021

Formula

T(n, k) = n!*binomial(n, k) - n! + 1.
From G. C. Greubel, Feb 09 2021: (Start)
T(n, k) = A196347(n, k) - n! + 1 = (-1)^k * A021012(n, k) - n! + 1.
Sum_{k=0..n} T(n, k) = 2^n * n! - (n+1)! + (n+1) = A000165(n) - (n+1)! + (n+1). (End)

Extensions

Edited by G. C. Greubel, Feb 09 2021

A253666 Triangle read by rows, T(n,k) = C(n,k)*n!/(floor(n/2)!)^2, n>=0, 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 6, 18, 18, 6, 6, 24, 36, 24, 6, 30, 150, 300, 300, 150, 30, 20, 120, 300, 400, 300, 120, 20, 140, 980, 2940, 4900, 4900, 2940, 980, 140, 70, 560, 1960, 3920, 4900, 3920, 1960, 560, 70, 630, 5670, 22680, 52920, 79380, 79380, 52920, 22680, 5670, 630
Offset: 0

Views

Author

Peter Luschny, Feb 01 2015

Keywords

Examples

			Triangle begins:
.   1;
.   1,    1;
.   2,    4,     2;
.   6,   18,    18,     6;
.   6,   24,    36,    24,     6;
.  30,  150,   300,   300,   150,    30;
.  20,  120,   300,   400,   300,   120,    20;
. 140,  980,  2940,  4900,  4900,  2940,   980,   140;
.  70,  560,  1960,  3920,  4900,  3920,  1960,   560,   70;
. 630, 5670, 22680, 52920, 79380, 79380, 52920, 22680, 5670, 630; etc.

Crossrefs

Cf. A021012, A196347, A002894.
Row sums are A253665.

Programs

  • Magma
    [Binomial(n,k)*Factorial(n)/Factorial(Floor(n/2))^2: k in [0..n], n in [0..10]]; // Bruno Berselli, Feb 02 2015
  • Maple
    T := (n,k) -> n!*binomial(n,k)/(iquo(n,2)!)^2:
seq(print(seq(T(n,k), k=0..n)), n=0..9);

Formula

T(n,k) = C(n,k)*A056040(k).
T(2*n,n) = C(2*n,n)^2.

A360283 a(n) = lcm({n! * binomial(n, k) for k = 0..n}).

Original entry on oeis.org

1, 1, 4, 18, 288, 1200, 43200, 529200, 11289600, 91445760, 9144576000, 92207808000, 13277924352000, 160283515392000, 2094371267788800, 58904191906560000, 15079473128079360000, 242109318556385280000, 78443419212268830720000, 1415903716781452394496000
Offset: 0

Views

Author

Peter Luschny, Feb 14 2023

Keywords

Crossrefs

Cf. A002397, A003418 (LCM), A005722, A021012, A196347.

Programs

  • Maple
    a := n -> ilcm(seq(n!*binomial(n, k), k=0..n)):
seq(a(n), n = 0..19);
  • Python
    from math import factorial, lcm
def A360283(n): return factorial(n)*lcm(*(i for i in range(1,n+2)))//(n+1) # Chai Wah Wu, Feb 15 2023

Formula

a(n) = n! * lcm({k for k = 1..n+1}) / (n+1) = n! * LCM(n + 1) / (n + 1).
a(n) / a(n-1) = n^2 if and only if n + 1 is prime, for n >= 1.
Showing 1-4 of 4 results.

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