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