cp's OEIS Frontend

A157268 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 A157268 #8 Feb 05 2022 02:32:05
%S A157268 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 A157268 5856,2413,182,1,1,373,8679,40337,40337,8679,373,1,1,756,29376,232726,
%U A157268 497066,232726,29376,756,1,1,1523,95668,1205968,4527078,4527078,1205968,95668,1523,1
%N A157268 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 A157268 G. C. Greubel, <a href="/A157268/b157268.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157268 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 A157268 T(n, n-k, m) = T(n, k, m).
%F A157268 T(n, 1, 1) = A101945(n-1), n >= 1. - _G. C. Greubel_, Feb 04 2022
%e A157268 Triangle begins as:
%e A157268   1;
%e A157268   1,    1;
%e A157268   1,    6,     1;
%e A157268   1,   17,    17,       1;
%e A157268   1,   40,   126,      40,       1;
%e A157268   1,   87,   606,     606,      87,       1;
%e A157268   1,  182,  2413,    5856,    2413,     182,       1;
%e A157268   1,  373,  8679,   40337,   40337,    8679,     373,     1;
%e A157268   1,  756, 29376,  232726,  497066,  232726,   29376,   756,    1;
%e A157268   1, 1523, 95668, 1205968, 4527078, 4527078, 1205968, 95668, 1523, 1;
%t A157268 f[n_,k_]:= If[k<=Floor[n/2], 2^k, 2^(n-k)];
%t A157268 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 A157268 Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 04 2022 *)
%o A157268 (Sage)
%o A157268 def f(n,k): return 2^k if (k <= n//2) else 2^(n-k)
%o A157268 @CachedFunction
%o A157268 def T(n,k,m):  # A157207
%o A157268     if (k==0 or k==n): return 1
%o A157268     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 A157268 flatten([[T(n,k,1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 04 2022
%Y A157268 Cf. A007318 (m=0), this sequence (m=1).
%Y A157268 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157272, A157273, A157274, A157275, A157277, A157278.
%Y A157268 Cf. A101945.
%K A157268 nonn,tabl
%O A157268 0,5
%A A157268 _Roger L. Bagula_, Feb 26 2009
%E A157268 Edited by _G. C. Greubel_, Feb 04 2022