A058298 -id:A058298 - cp's OEIS frontend

A077012 Triangle in which n-th row contains all possible products of n-1 of the first n natural numbers in ascending order.

1, 1, 2, 2, 3, 6, 6, 8, 12, 24, 24, 30, 40, 60, 120, 120, 144, 180, 240, 360, 720, 720, 840, 1008, 1260, 1680, 2520, 5040, 5040, 5760, 6720, 8064, 10080, 13440, 20160, 40320, 40320, 45360, 51840, 60480, 72576, 90720, 120960, 181440, 362880

Offset: 1

Amarnath Murthy, Oct 26 2002

Row products of A137853. - Mats Granvik, Jun 24 2009

			Triangle begins:
     1;
     1,    2;
     2,    3,    6;
     6,    8,   12,   24;
    24,   30,   40,   60,   120;
   120,  144,  180,  240,   360,   720;
   720,  840, 1008, 1260,  1680,  2520,  5040;
  5040, 5760, 6720, 8064, 10080, 13440, 20160, 40320;
  ...

Row sums give A000254.

Cf. A058298, A078921, A137853, A196533.

A077012_row := proc(n) local k; seq(n!/(n-k),k=0..n-1) end:
seq(print(A077012_row(n)),n=1..9); # Peter Luschny, Mar 21 2011

Flatten[Table[n!/(n-k),{n,10},{k,0,n-1}]] (* Harvey P. Dale, Dec 25 2011 *)

E.g.f.: -log(1-x)/(1-y*x). - Vladeta Jovovic, Feb 07 2003

Sum_{n>=1} Sum_{k=0..n-1} 1/T(n, k) = 3*e/2 (= A196533 / 10). - Amiram Eldar, Jun 29 2025

More terms from Sascha Kurz, Jan 26 2003

A156047 Triangle read by rows: T(n, k) = (n+1)!*(1/k + 1/(n-k+1)).

Original entry on oeis.org

4, 9, 9, 32, 24, 32, 150, 100, 100, 150, 864, 540, 480, 540, 864, 5880, 3528, 2940, 2940, 3528, 5880, 46080, 26880, 21504, 20160, 21504, 26880, 46080, 408240, 233280, 181440, 163296, 163296, 181440, 233280, 408240, 4032000, 2268000, 1728000, 1512000, 1451520, 1512000, 1728000, 2268000, 4032000

Offset: 1

Roger L. Bagula, Feb 02 2009

Row sums are (n+1)*A052517(n+2) = {4, 18, 88, 500, 3288, 24696, 209088, 1972512, 20531520, ...}.

			Triangle begins as:
      4;
      9,     9;
     32,    24,    32;
    150,   100,   100,   150;
    864,   540,   480,   540,   864;
   5880,  3528,  2940,  2940,  3528,  5880;
  46080, 26880, 21504, 20160, 21504, 26880, 46080;

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A001008, A002805, A052517, A058298.

Flat(List([1..10], n-> List([1..n], k-> (n+1)*Factorial(n+1)/(k*(n-k+1)) ))); # G. C. Greubel, Dec 02 2019

[(n+1)*Factorial(n+1)/(k*(n-k+1)): k in [1..n], n in [1..10]]; // G. C. Greubel, Dec 02 2019

seq(seq( (n+1)*(n+1)!/(k*(n-k+1)), k=1..n), n=1..10); # G. C. Greubel, Dec 02 2019

Table[(n+1)*(n+1)!/(k*(n-k+1)), {n,10}, {k,n}]//Flatten (* modified by G. C. Greubel, Dec 02 2019 *)

T(n,k) = (n+1)*(n+1)!/(k*(n-k+1)); \\ G. C. Greubel, Dec 02 2019

[[(n+1)*factorial(n+1)/(k*(n-k+1)) for k in (1..n)] for n in (1..10)] # G. C. Greubel, Dec 02 2019

T(n, k) = (n+1)*(n+1)!/(k*(n-k+1)).

Sum_{k=1..n} T(n,k) = 2*(n+1)!*H(n), where H(n) is the harmonic number. - G. C. Greubel, Dec 02 2019

Offset changed by G. C. Greubel, Dec 02 2019

cp's OEIS Frontend

A077012 Triangle in which n-th row contains all possible products of n-1 of the first n natural numbers in ascending order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions