A159623 - OEIS

%I #13 Nov 28 2021 03:04:11

%S 1,1,1,1,2,1,1,3,3,1,1,4,12,4,1,1,5,20,20,5,1,1,6,30,120,30,6,1,1,7,

%T 42,210,210,42,7,1,1,8,56,336,1680,336,56,8,1,1,9,72,504,3024,3024,

%U 504,72,9,1,1,10,90,720,5040,30240,5040,720,90,10,1

%N Triangle read by rows: T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 1.

%C The row sums are: {1, 2, 4, 8, 22, 52, 194, 520, 2482, 7220, 41962,...}.

%H G. C. Greubel, <a href="/A159623/b159623.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = n!*q^k/(n-k)! if floor(n/2) > k-1 otherwise n!*q^(n-k)/k!, with q = 1.

%F T(n, n-k) = T(n, k).

%F T(2*n, n) = A001813(n). - _G. C. Greubel_, Nov 28 2021

%e Triangle begins as:

%e   1;

%e   1,  1;

%e   1,  2,  1;

%e   1,  3,  3,   1;

%e   1,  4, 12,   4,    1;

%e   1,  5, 20,  20,    5,     1;

%e   1,  6, 30, 120,   30,     6,    1;

%e   1,  7, 42, 210,  210,    42,    7,   1;

%e   1,  8, 56, 336, 1680,   336,   56,   8,  1;

%e   1,  9, 72, 504, 3024,  3024,  504,  72,  9,  1;

%e   1, 10, 90, 720, 5040, 30240, 5040, 720, 90, 10,  1;

%t T[n_, k_, q_]:= If[Floor[n/2]>=k, n!*q^k/(n-k)!, n!*q^(n-k)/k!];

%t Table[T[n, k, 1], {n,0,12}, {k,0,n}]//Flatten

%o (Sage)

%o f=factorial

%o def T(n,k,q): return f(n)*q^k/f(n-k) if ((n//2)>k-1) else f(n)*q^(n-k)/f(k)

%o flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Nov 28 2021

%Y Cf. this sequence (q=1), A174376 (q=2), A174377 (q=3), A174378 (q=4).

%K nonn,tabl,easy

%O 0,5

%A _Roger L. Bagula_, Apr 17 2009

%E Edited by _N. J. A. Sloane_, Apr 17 2009