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%I #6 Feb 05 2022 06:43:52
%S 1,1,1,1,17,1,1,84,84,1,1,355,1431,355,1,1,1442,14827,14827,1442,1,1,
%T 5793,127860,326591,127860,5793,1,1,23200,1009338,5239457,5239457,
%U 1009338,23200,1,1,92831,7593061,71229038,145043839,71229038,7593061,92831,1
%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*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 3, read by rows.
%H G. C. Greubel, <a href="/A157274/b157274.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*f(n,k)*T(n-2, k-1, m) with T(n, 0, m) = T(n, n, m) = 1, f(n, k) = 2*k+1 if k <= floor(n/2) otherwise 2*(n-k)+1, and m = 3.
%F T(n, n-k, m) = T(n, k, m).
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 17, 1;
%e 1, 84, 84, 1;
%e 1, 355, 1431, 355, 1;
%e 1, 1442, 14827, 14827, 1442, 1;
%e 1, 5793, 127860, 326591, 127860, 5793, 1;
%e 1, 23200, 1009338, 5239457, 5239457, 1009338, 23200, 1;
%e 1, 92831, 7593061, 71229038, 145043839, 71229038, 7593061, 92831, 1;
%t f[n_,k_]:= If[k<=Floor[n/2], 2*k+1, 2*(n-k)+1];
%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*f[n,k]*T[n-2,k-1,m]];
%t Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 05 2022 *)
%o (Sage)
%o def f(n,k): return 2*k+1 if (k <= n//2) else 2*(n-k)+1
%o @CachedFunction
%o def T(n,k,m): # A157207
%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*f(n,k)*T(n-2,k-1,m)
%o flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 05 2022
%Y Cf. A007318 (m=0), A157272 (m=1), A157273 (m=2), this sequence (m=3).
%Y Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157275, A157277, A157278.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 26 2009
%E Edited by _G. C. Greubel_, Feb 05 2022
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