cp's OEIS Frontend

A157208 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) = k if k <= floor(n/2) otherwise n-k, and m = 2, read by rows.

%I A157208 #7 Jan 10 2022 18:20:22
%S A157208 1,1,1,1,8,1,1,31,31,1,1,102,342,102,1,1,317,2548,2548,317,1,1,964,
%T A157208 16001,37724,16001,964,1,1,2907,91877,423365,423365,91877,2907,1,1,
%U A157208 8738,501032,4070208,7922362,4070208,501032,8738,1,1,26233,2647858,35556134,119460466,119460466,35556134,2647858,26233,1
%N A157208 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) = k if k <= floor(n/2) otherwise n-k, and m = 2, read by rows.
%H A157208 G. C. Greubel, <a href="/A157208/b157208.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157208 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) = k if k <= floor(n/2) otherwise n-k, and m = 2.
%F A157208 T(n, n-k, m) = T(n, k, m).
%e A157208 Triangle begins as:
%e A157208   1;
%e A157208   1,    1;
%e A157208   1,    8,      1;
%e A157208   1,   31,     31,       1;
%e A157208   1,  102,    342,     102,       1;
%e A157208   1,  317,   2548,    2548,     317,       1;
%e A157208   1,  964,  16001,   37724,   16001,     964,      1;
%e A157208   1, 2907,  91877,  423365,  423365,   91877,   2907,    1;
%e A157208   1, 8738, 501032, 4070208, 7922362, 4070208, 501032, 8738, 1;
%t A157208 f[n_,k_]:= If[k<=Floor[n/2], k, n-k];
%t A157208 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 A157208 Table[T[n,k,2], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 10 2022 *)
%o A157208 (Sage)
%o A157208 def f(n,k): return k if (k <= n//2) else n-k
%o A157208 @CachedFunction
%o A157208 def T(n,k,m):  # A157208
%o A157208     if (k==0 or k==n): return 1
%o A157208     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 A157208 flatten([[T(n,k,2) for k in (0..n)] for n in (0..20)]) # _G. C. Greubel_, Jan 10 2022
%Y A157208 Cf. A007318 (m=0), A157207 (m=1), this sequence (m=2), A157209 (m=3).
%Y A157208 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
%K A157208 nonn,tabl
%O A157208 0,5
%A A157208 _Roger L. Bagula_, Feb 25 2009
%E A157208 Edited by _G. C. Greubel_, Jan 10 2022