cp's OEIS Frontend

A157209 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 = 3, read by rows.

%I A157209 #5 Jan 10 2022 18:21:24
%S A157209 1,1,1,1,11,1,1,54,54,1,1,229,822,229,1,1,932,8368,8368,932,1,1,3747,
%T A157209 72066,174758,72066,3747,1,1,15010,570006,2759750,2759750,570006,
%U A157209 15010,1,1,60065,4297714,37366190,73850596,37366190,4297714,60065,1,1,240288,31495488,460448520,1591033788,1591033788,460448520,31495488,240288,1
%N A157209 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 = 3, read by rows.
%H A157209 G. C. Greubel, <a href="/A157209/b157209.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157209 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 = 3.
%F A157209 T(n, n-k, m) = T(n, k, m).
%e A157209 Triangle begins as:
%e A157209   1;
%e A157209   1,     1;
%e A157209   1,    11,       1;
%e A157209   1,    54,      54,        1;
%e A157209   1,   229,     822,      229,        1;
%e A157209   1,   932,    8368,     8368,      932,        1;
%e A157209   1,  3747,   72066,   174758,    72066,     3747,       1;
%e A157209   1, 15010,  570006,  2759750,  2759750,   570006,   15010,     1;
%e A157209   1, 60065, 4297714, 37366190, 73850596, 37366190, 4297714, 60065, 1;
%t A157209 f[n_,k_]:= If[k<=Floor[n/2], k, n-k];
%t A157209 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 A157209 Table[T[n,k,3], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 10 2022 *)
%o A157209 (Sage)
%o A157209 def f(n,k): return k if (k <= n//2) else n-k
%o A157209 @CachedFunction
%o A157209 def T(n,k,m):  # A157209
%o A157209     if (k==0 or k==n): return 1
%o A157209     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 A157209 flatten([[T(n,k,3) for k in (0..n)] for n in (0..20)]) # _G. C. Greubel_, Jan 10 2022
%Y A157209 Cf. A007318 (m=0), A157207 (m=1), A157208 (m=2), this sequence (m=3).
%Y A157209 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157210, A157211, A157212, A157268, A157272, A157273, A157274, A157275.
%K A157209 nonn,tabl
%O A157209 0,5
%A A157209 _Roger L. Bagula_, Feb 25 2009
%E A157209 Edited by _G. C. Greubel_, Jan 10 2022