cp's OEIS Frontend

A157154 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 3, read by rows.

%I A157154 #10 Jan 10 2022 13:28:22
%S A157154 1,1,1,1,5,1,1,21,21,1,1,85,234,85,1,1,341,2110,2110,341,1,1,1365,
%T A157154 17163,35882,17163,1365,1,1,5461,131751,505979,505979,131751,5461,1,1,
%U A157154 21845,976876,6395471,11433118,6395471,976876,21845,1,1,87381,7089360,75400800,220599330,220599330,75400800,7089360,87381,1
%N A157154 Triangle T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 3, read by rows.
%H A157154 G. C. Greubel, <a href="/A157154/b157154.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157154 T(n, k, m) = (m*(n-k) + 1)*T(n-1, k-1, m) + (m*k + 1)*T(n-1, k, m) - m*k*(n-k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1 and m = 1.
%F A157154 T(n, n-k, m) = T(n, k, m).
%F A157154 T(n, 1, 3) = A002450(n). - _G. C. Greubel_, Jan 10 2022
%e A157154 Triangle begins as:
%e A157154   1;
%e A157154   1,     1;
%e A157154   1,     5,       1;
%e A157154   1,    21,      21,        1;
%e A157154   1,    85,     234,       85,         1;
%e A157154   1,   341,    2110,     2110,       341,         1;
%e A157154   1,  1365,   17163,    35882,     17163,      1365,        1;
%e A157154   1,  5461,  131751,   505979,    505979,    131751,     5461,       1;
%e A157154   1, 21845,  976876,  6395471,  11433118,   6395471,   976876,   21845,     1;
%e A157154   1, 87381, 7089360, 75400800, 220599330, 220599330, 75400800, 7089360, 87381, 1;
%t A157154 T[n_, k_, m_]:= T[n, k, m]= If[k==0 || k==n, 1, (m*(n-k)+1)*T[n-1,k-1,m] + (m*k+1)*T[n-1,k,m] - m*k*(n-k)*T[n-2,k-1,m]];
%t A157154 Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 10 2022 *)
%o A157154 (Sage)
%o A157154 @CachedFunction
%o A157154 def T(n,k,m):  # A157154
%o A157154     if (k==0 or k==n): return 1
%o A157154     else: return (m*(n-k) +1)*T(n-1,k-1,m) + (m*k+1)*T(n-1,k,m) - m*k*(n-k)*T(n-2,k-1,m)
%o A157154 flatten([[T(n,k,3) for k in (0..n)] for n in (0..20)]) # _G. C. Greubel_, Jan 10 2022
%Y A157154 Cf. A007318 (m=0), A157152 (m=1), A157153 (m=2), this sequence (m=3), A157155 (m=4), A157156 (m=5).
%Y A157154 Cf. A157147, A157148, A157149, A157150, A157151, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
%Y A157154 Cf. A002450.
%K A157154 nonn,tabl,easy
%O A157154 0,5
%A A157154 _Roger L. Bagula_, Feb 24 2009
%E A157154 Edited by _G. C. Greubel_, Jan 10 2022