cp's OEIS Frontend

A157275 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 if k <= floor(n/2) otherwise 2*(n-k), and m = 1, read by rows.

%I A157275 #6 Feb 06 2022 02:06:31
%S A157275 1,1,1,1,6,1,1,17,17,1,1,40,126,40,1,1,87,606,606,87,1,1,182,2413,
%T A157275 5604,2413,182,1,1,373,8679,38117,38117,8679,373,1,1,756,29376,219020,
%U A157275 426002,219020,29376,756,1,1,1523,95668,1133786,3749066,3749066,1133786,95668,1523,1
%N A157275 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 if k <= floor(n/2) otherwise 2*(n-k), and m = 1, read by rows.
%H A157275 G. C. Greubel, <a href="/A157275/b157275.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157275 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 if k <= floor(n/2) otherwise 2*(n-k), and m = 1.
%F A157275 T(n, n-k, m) = T(n, k, m).
%F A157275 T(n, 1, 1) = A101945(n-1), for n >= 1. - _G. C. Greubel_, Feb 05 2022
%e A157275 Triangle begins as:
%e A157275   1;
%e A157275   1,    1;
%e A157275   1,    6,     1;
%e A157275   1,   17,    17,       1;
%e A157275   1,   40,   126,      40,       1;
%e A157275   1,   87,   606,     606,      87,       1;
%e A157275   1,  182,  2413,    5604,    2413,     182,       1;
%e A157275   1,  373,  8679,   38117,   38117,    8679,     373,     1;
%e A157275   1,  756, 29376,  219020,  426002,  219020,   29376,   756,    1;
%e A157275   1, 1523, 95668, 1133786, 3749066, 3749066, 1133786, 95668, 1523, 1;
%t A157275 f[n_,k_]:= If[k<=Floor[n/2], 2*k, 2*(n-k)];
%t A157275 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 A157275 Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 05 2022 *)
%o A157275 (Sage)
%o A157275 def f(n,k): return 2*k if (k <= n//2) else 2*(n-k)
%o A157275 @CachedFunction
%o A157275 def T(n,k,m):  # A157275
%o A157275     if (k==0 or k==n): return 1
%o A157275     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 A157275 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 05 2022
%Y A157275 Cf. A007318 (m=0), this sequence (m=1), A157277 (m=2), A157278 (m=3).
%Y A157275 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274.
%Y A157275 Cf. A101945.
%K A157275 nonn,tabl
%O A157275 0,5
%A A157275 _Roger L. Bagula_, Feb 26 2009
%E A157275 Edited by _G. C. Greubel_, Feb 05 2022