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A051683 Triangle read by rows: T(n,k) = n!*k.

%I A051683 #69 Jun 25 2025 16:46:56
%S A051683 1,2,4,6,12,18,24,48,72,96,120,240,360,480,600,720,1440,2160,2880,
%T A051683 3600,4320,5040,10080,15120,20160,25200,30240,35280,40320,80640,
%U A051683 120960,161280,201600,241920,282240,322560,362880,725760,1088640,1451520,1814400,2177280,2540160,2903040,3265920
%N A051683 Triangle read by rows: T(n,k) = n!*k.
%C A051683 Numbers with only one nonzero digit when written in factorial base. - _Franklin T. Adams-Watters_, Nov 28 2011
%C A051683 In other words, numbers m such that A034968(m) = A099563(m). - _Antti Karttunen_, Jul 02 2013
%C A051683 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 A051683 Reinhard Zumkeller, <a href="/A051683/b051683.txt">Rows n=1..150 of triangle, flattened</a>
%H A051683 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 A051683 T(n,k) = A000142(A002024(n)) * A002260(n,k) = A002024(n)! * A002260(n,k) - _Antti Karttunen_, Jul 02 2013
%F A051683 Sum_{n>=1} 1/a(n) = e * (gamma - Ei(-1)) = A347952. - _Amiram Eldar_, Oct 13 2024
%e A051683 Table begins
%e A051683    1;
%e A051683    2,  4;
%e A051683    6, 12, 18;
%e A051683   24, 48, 72, 96; ...
%t A051683 T[n_, k_] := n!*k; Flatten[Table[T[n, k], {n, 9}, {k, n}]] (* _Jean-François Alcover_, Apr 22 2011 *)
%o A051683 (Haskell)
%o A051683 a051683 n k = a051683_tabl !! (n-1) !! (k-1)
%o A051683 a051683_row n = a051683_tabl !! (n-1)
%o A051683 a051683_tabl = map fst $ iterate f ([1], 2) where
%o A051683    f (row, n) = (row' ++ [head row' + last row'], n + 1) where
%o A051683      row' = map (* n) row
%o A051683 -- _Reinhard Zumkeller_, Mar 09 2012
%o A051683 (Scheme) (define (A051683 n) (* (A000142 (A002024 n)) (A002260 n))) ;; _Antti Karttunen_, Jul 02 2013
%o A051683 (Magma) [[Factorial(n)*k: k in [1..n]]: n in [1..15]]; // _Vincenzo Librandi_, Jun 15 2015
%o A051683 (PARI) for(n=1,10, for(k=1,n, print1(n!*k, ", "))) \\ _G. C. Greubel_, Mar 27 2018
%o A051683 (Python)
%o A051683 from math import isqrt, factorial, comb
%o A051683 def A051683(n): return factorial(a:=(m:=isqrt(k:=n<<1))+(k>m*(m+1)))*(n-comb(a,2)) # _Chai Wah Wu_, Jun 25 2025
%Y A051683 Cf. A000142, A002024, A002260, row sums give A001286(n+1).
%Y A051683 Cf. A001563, A007623, A034968, A099563, A200748, A162608, A347952.
%K A051683 easy,nice,nonn,tabl
%O A051683 1,2
%A A051683 Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)