cp's OEIS Frontend

A157211 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 A157211 #8 Jan 10 2022 18:13:39
%S A157211 1,1,1,1,4,1,1,15,15,1,1,50,134,50,1,1,157,960,960,157,1,1,480,6013,
%T A157211 12636,6013,480,1,1,1451,34717,136809,136809,34717,1451,1,1,4366,
%U A157211 190528,1303472,2361474,1303472,190528,4366,1,1,13113,1012326,11392866,34496986,34496986,11392866,1012326,13113,1
%N A157211 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 A157211 G. C. Greubel, <a href="/A157211/b157211.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157211 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 A157211 T(n, n-k, m) = T(n, k, m).
%F A157211 T(n, 1, 2) = A132308(n-1). - _G. C. Greubel_, Jan 10 2022
%e A157211 Triangle begins as:
%e A157211   1;
%e A157211   1,     1;
%e A157211   1,     4,       1;
%e A157211   1,    15,      15,        1;
%e A157211   1,    50,     134,       50,        1;
%e A157211   1,   157,     960,      960,      157,        1;
%e A157211   1,   480,    6013,    12636,     6013,      480,        1;
%e A157211   1,  1451,   34717,   136809,   136809,    34717,     1451,       1;
%e A157211   1,  4366,  190528,  1303472,  2361474,  1303472,   190528,    4366,     1;
%e A157211   1, 13113, 1012326, 11392866, 34496986, 34496986, 11392866, 1012326, 13113, 1;
%t A157211 f[n_,k_]:= If[k<=Floor[n/2], k, n-k];
%t A157211 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 A157211 Table[T[n,k,2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 10 2022 *)
%o A157211 (Sage)
%o A157211 def f(n,k): return k if (k <= n//2) else n-k
%o A157211 @CachedFunction
%o A157211 def T(n,k,m):  # A157211
%o A157211     if (k==0 or k==n): return 1
%o A157211     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 A157211 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 10 2022
%Y A157211 Cf. A007318 (m=0), A157210 (m=1), this sequence (m=2), A157212 (m=3).
%Y A157211 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157268, A157272, A157273, A157274, A157275.
%Y A157211 Cf. A132308.
%K A157211 nonn,tabl
%O A157211 0,5
%A A157211 _Roger L. Bagula_, Feb 25 2009
%E A157211 Edited by _G. C. Greubel_, Jan 10 2022