cp's OEIS Frontend

A157272 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 = 1, read by rows.

%I A157272 #6 Feb 05 2022 02:31:57
%S A157272 1,1,1,1,7,1,1,20,20,1,1,47,155,47,1,1,102,753,753,102,1,1,213,3004,
%T A157272 7109,3004,213,1,1,436,10800,48727,48727,10800,436,1,1,883,36517,
%U A157272 280736,551251,280736,36517,883,1,1,1778,118795,1454163,4879214,4879214,1454163,118795,1778,1
%N A157272 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 = 1, read by rows.
%H A157272 G. C. Greubel, <a href="/A157272/b157272.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157272 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 = 1.
%F A157272 T(n, n-k, m) = T(n, k, m).
%e A157272 Triangle begins as:
%e A157272   1;
%e A157272   1,    1;
%e A157272   1,    7,      1;
%e A157272   1,   20,     20,       1;
%e A157272   1,   47,    155,      47,       1;
%e A157272   1,  102,    753,     753,     102,       1;
%e A157272   1,  213,   3004,    7109,    3004,     213,       1;
%e A157272   1,  436,  10800,   48727,   48727,   10800,     436,      1;
%e A157272   1,  883,  36517,  280736,  551251,  280736,   36517,    883,    1;
%e A157272   1, 1778, 118795, 1454163, 4879214, 4879214, 1454163, 118795, 1778, 1;
%t A157272 f[n_,k_]:= If[k<=Floor[n/2], 2*k+1, 2*(n-k)+1];
%t A157272 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 A157272 Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 04 2022 *)
%o A157272 (Sage)
%o A157272 def f(n,k): return 2*k+1 if (k <= n//2) else 2*(n-k)+1
%o A157272 @CachedFunction
%o A157272 def T(n,k,m):  # A157207
%o A157272     if (k==0 or k==n): return 1
%o A157272     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 A157272 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 04 2022
%Y A157272 Cf. A007318 (m=0), this sequence (m=1), A157273 (m=2), A157274 (m=3).
%Y A157272 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157275, A157277, A157278.
%K A157272 nonn,tabl
%O A157272 0,5
%A A157272 _Roger L. Bagula_, Feb 26 2009
%E A157272 Edited by _G. C. Greubel_, Feb 04 2022