A158116 - OEIS

%I #32 Sep 08 2022 08:45:42

%S 1,1,1,1,6,1,1,36,36,1,1,216,1296,216,1,1,1296,46656,46656,1296,1,1,

%T 7776,1679616,10077696,1679616,7776,1,1,46656,60466176,2176782336,

%U 2176782336,60466176,46656,1,1,279936,2176782336,470184984576,2821109907456,470184984576,2176782336,279936,1

%N Triangle T(n,k) = 6^(k*(n-k)), read by rows.

%H G. C. Greubel, <a href="/A158116/b158116.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n,k) = 6^(k*(n-k)). - _Tom Edgar_, Feb 20 2014

%F T(n,k) = (1/n)*(6^(n-k)*k*T(n-1,k-1) + 6^k*(n-k)*T(n-1,k)). - _Tom Edgar_, Feb 20 2014

%F From _G. C. Greubel_, Jun 30 2021: (Start)

%F T(n, k, m) = (m+2)^(k*(n-k)) with m = 4.

%F T(n, k, q) = binomial(2*q, 2)^(k*(n-k)) with q = 2. (End)

%e Triangle starts:

%e   1;

%e   1,     1;

%e   1,     6,        1;

%e   1,    36,       36,          1;

%e   1,   216,     1296,        216,          1;

%e   1,  1296,    46656,      46656,       1296,        1;

%e   1,  7776,  1679616,   10077696,    1679616,     7776,     1;

%e   1, 46656, 60466176, 2176782336, 2176782336, 60466176, 46656, 1;

%t With[{m=4}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* _G. C. Greubel_, Jun 30 2021 *)

%o (PARI) T(n,k) = 6^(k*(n-k));

%o for (n=0,11,for (k=0,n, print1(T(n,k),", "));print();); \\ _Joerg Arndt_, Feb 21 2014

%o (Magma) [6^(k*(n-k)): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 30 2021

%o (Sage) flatten([[6^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 30 2021

%Y Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), this sequence (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).

%Y Cf. this sequence (q=2), A176639 (q=3), A176641 (q=4).

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Mar 12 2009

%E Overall edit and new name by _Tom Edgar_ and _Joerg Arndt_, Feb 21 2014