A051683 - OEIS

%I #62 Oct 13 2024 11:17:04

%S 1,2,4,6,12,18,24,48,72,96,120,240,360,480,600,720,1440,2160,2880,

%T 3600,4320,5040,10080,15120,20160,25200,30240,35280,40320,80640,

%U 120960,161280,201600,241920,282240,322560,362880,725760,1088640,1451520,1814400,2177280,2540160,2903040,3265920

%N Triangle read by rows: T(n,k) = n!*k.

%C Numbers with only one nonzero digit when written in factorial base. - _Franklin T. Adams-Watters_, Nov 28 2011

%C In other words, numbers m such that A034968(m) = A099563(m). - _Antti Karttunen_, Jul 02 2013

%C When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the right within an interval. The subsequence A001563 denotes the circular shifts that start with the first element. Compare A211370 for circular shifts to the left. - _Tilman Piesk_, Apr 29 2017

%H Reinhard Zumkeller, <a href="/A051683/b051683.txt">Rows n=1..150 of triangle, flattened</a>

%H Tilman Piesk, <a href="https://commons.wikimedia.org/wiki/File:Array_of_permutations;_circular_shift,_right.svg">Circular shifts to the right</a> (<a href="https://en.wikiversity.org/wiki/Arrays_of_permutations#arrays4">Arrays of permutations</a>).

%F T(n,k) = A000142(A002024(n)) * A002260(n,k) = A002024(n)! * A002260(n,k) - _Antti Karttunen_, Jul 02 2013

%F Sum_{n>=1} 1/a(n) = e * (gamma - Ei(-1)) = A347952. - _Amiram Eldar_, Oct 13 2024

%e Table begins

%e    1;

%e    2,  4;

%e    6, 12, 18;

%e   24, 48, 72, 96; ...

%t T[n_, k_] := n!*k; Flatten[Table[T[n, k], {n, 9}, {k, n}]] (* _Jean-François Alcover_, Apr 22 2011 *)

%o (Haskell)

%o a051683 n k = a051683_tabl !! (n-1) !! (k-1)

%o a051683_row n = a051683_tabl !! (n-1)

%o a051683_tabl = map fst $ iterate f ([1], 2) where

%o    f (row, n) = (row' ++ [head row' + last row'], n + 1) where

%o      row' = map (* n) row

%o -- _Reinhard Zumkeller_, Mar 09 2012

%o (Scheme): (define (A051683 n) (* (A000142 (A002024 n)) (A002260 n))) -- _Antti Karttunen_, Jul 02 2013

%o (Magma) [[Factorial(n)*k: k in [1..n]]: n in [1..15]]; // _Vincenzo Librandi_, Jun 15 2015

%o (PARI) for(n=1,10, for(k=1,n, print1(n!*k, ", "))) \\ _G. C. Greubel_, Mar 27 2018

%Y Cf. A000142, A002024, A002260, row sums give A001286(n+1).

%Y Cf. A001563, A007623, A034968, A099563, A200748, A162608, A347952.

%K easy,nice,nonn,tabl

%O 1,2

%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)