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%I #16 Jan 10 2022 03:07:05
%S 1,1,1,1,17,1,1,123,123,1,1,769,3046,769,1,1,4655,49500,49500,4655,1,
%T 1,27981,673015,1721070,673015,27981,1,1,167947,8363421,44640435,
%U 44640435,8363421,167947,1,1,1007753,98882848,982172031,2012583870,982172031,98882848,1007753,1
%N 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 = 5, read by rows.
%H G. C. Greubel, <a href="/A157151/b157151.txt">Rows n = 0..50 of the triangle, flattened</a>
%F 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 = 5.
%F T(n, n-k, 5) = T(n, k, 5).
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 17, 1;
%e 1, 123, 123, 1;
%e 1, 769, 3046, 769, 1;
%e 1, 4655, 49500, 49500, 4655, 1;
%e 1, 27981, 673015, 1721070, 673015, 27981, 1;
%e 1, 167947, 8363421, 44640435, 44640435, 8363421, 167947, 1;
%e 1, 1007753, 98882848, 982172031, 2012583870, 982172031, 98882848, 1007753, 1;
%p A157151:= proc(n, k)
%p if k<0 or n<k then 0;
%p elif k=0 or k=n then 1;
%p else (5*n-5*k+1)*procname(n-1, k-1) + (5*k+1)*procname(n-1, k) + 5*k*(n-k)*procname(n-2, k-1);
%p end if; end proc;
%p seq(seq(A157151(n, k), k=0..n), n=0..10); # _R. J. Mathar_, Feb 06 2015
%t 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 Table[T[n,k,5], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 09 2022 *)
%o (Sage)
%o def T(n,k,m): # A157147
%o if (k==0 or k==n): return 1
%o 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 flatten([[T(n,k,5) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Jan 09 2022
%Y Cf. A007318 (m=0), A157147 (m=1), A157148 (m=2), A157149 (m=3), A157150 (m=4), this sequence (m=5).
%Y Cf. A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
%K nonn,tabl,easy
%O 0,5
%A _Roger L. Bagula_, Feb 24 2009
%E Edited by _G. C. Greubel_, Jan 09 2022
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