|
%I #6 Feb 05 2022 06:44:14
%S 1,1,1,1,12,1,1,47,47,1,1,154,590,154,1,1,477,4498,4498,477,1,1,1448,
%T 28323,71232,28323,1448,1,1,4363,162313,816503,816503,162313,4363,1,1,
%U 13110,882764,7897486,15979230,7897486,882764,13110,1,1,39353,4654100,69030716,245382470,245382470,69030716,4654100,39353,1
%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 = 2, read by rows.
%H G. C. Greubel, <a href="/A157273/b157273.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 = 2.
%F T(n, n-k, m) = T(n, k, m).
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 12, 1;
%e 1, 47, 47, 1;
%e 1, 154, 590, 154, 1;
%e 1, 477, 4498, 4498, 477, 1;
%e 1, 1448, 28323, 71232, 28323, 1448, 1;
%e 1, 4363, 162313, 816503, 816503, 162313, 4363, 1;
%e 1, 13110, 882764, 7897486, 15979230, 7897486, 882764, 13110, 1;
%e 1, 39353, 4654100, 69030716, 245382470, 245382470, 69030716, 4654100, 39353, 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,2], {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,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 05 2022
%Y Cf. A007318 (m=0), A157272 (m=1), this sequence (m=2), A157274 (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
|
