A156047 - OEIS

%I #10 Sep 08 2022 08:45:41

%S 4,9,9,32,24,32,150,100,100,150,864,540,480,540,864,5880,3528,2940,

%T 2940,3528,5880,46080,26880,21504,20160,21504,26880,46080,408240,

%U 233280,181440,163296,163296,181440,233280,408240,4032000,2268000,1728000,1512000,1451520,1512000,1728000,2268000,4032000

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

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

%H G. C. Greubel, <a href="/A156047/b156047.txt">Rows n = 1..100 of triangle, flattened</a>

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

%F 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

%e Triangle begins as:

%e       4;

%e       9,     9;

%e      32,    24,    32;

%e     150,   100,   100,   150;

%e     864,   540,   480,   540,   864;

%e    5880,  3528,  2940,  2940,  3528,  5880;

%e   46080, 26880, 21504, 20160, 21504, 26880, 46080;

%p seq(seq( (n+1)*(n+1)!/(k*(n-k+1)), k=1..n), n=1..10); # _G. C. Greubel_, Dec 02 2019

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

%o (PARI) T(n,k) = (n+1)*(n+1)!/(k*(n-k+1)); \\ _G. C. Greubel_, Dec 02 2019

%o (Magma) [(n+1)*Factorial(n+1)/(k*(n-k+1)): k in [1..n], n in [1..10]]; // _G. C. Greubel_, Dec 02 2019

%o (Sage) [[(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

%o (GAP) Flat(List([1..10], n-> List([1..n], k-> (n+1)*Factorial(n+1)/(k*(n-k+1)) ))); # _G. C. Greubel_, Dec 02 2019

%Y Cf. A001008, A002805, A052517, A058298.

%K nonn,tabl,easy

%O 1,1

%A _Roger L. Bagula_, Feb 02 2009

%E Offset changed by _G. C. Greubel_, Dec 02 2019