cp's OEIS Frontend

A157210 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 = 1, read by rows.

%I A157210 #8 Jan 11 2022 02:19:01
%S A157210 1,1,1,1,3,1,1,8,8,1,1,19,42,19,1,1,42,186,186,42,1,1,89,730,1362,730,
%T A157210 89,1,1,184,2640,8540,8540,2640,184,1,1,375,9030,47810,79952,47810,
%U A157210 9030,375,1,1,758,29722,246530,652460,652460,246530,29722,758,1
%N A157210 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 = 1, read by rows.
%H A157210 G. C. Greubel, <a href="/A157210/b157210.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157210 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 = 1.
%F A157210 T(n, n-k, m) = T(n, k, m).
%F A157210 T(n, 1, 1) = A079583(n-1). - _G. C. Greubel_, Jan 10 2022
%e A157210 Triangle begins as:
%e A157210   1;
%e A157210   1,    1;
%e A157210   1,    3,     1;
%e A157210   1,    8,     8,       1;
%e A157210   1,   19,    42,      19,       1;
%e A157210   1,   42,   186,     186,      42,       1;
%e A157210   1,   89,   730,    1362,     730,      89,       1;
%e A157210   1,  184,  2640,    8540,    8540,    2640,     184,       1;
%e A157210   1,  375,  9030,   47810,   79952,   47810,    9030,     375,     1;
%e A157210   1,  758, 29722,  246530,  652460,  652460,  246530,   29722,   758,    1;
%e A157210   1, 1525, 95238, 1196806, 4796770, 7429760, 4796770, 1196806, 95238, 1525, 1;
%t A157210 f[n_,k_]:= If[k<=Floor[n/2], k, n-k];
%t A157210 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 A157210 Table[T[n,k,1], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 10 2022 *)
%o A157210 (Sage)
%o A157210 def f(n,k): return k if (k <= n//2) else n-k
%o A157210 @CachedFunction
%o A157210 def T(n,k,m):  # A157210
%o A157210     if (k==0 or k==n): return 1
%o A157210     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 A157210 flatten([[T(n,k,1) for k in (0..n)] for n in (0..20)]) # _G. C. Greubel_, Jan 10 2022
%Y A157210 Cf. A007318 (m=0), this sequence (m=1), A157211 (m=2), A157212 (m=3).
%Y A157210 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157268, A157272, A157273, A157274, A157275.
%Y A157210 Cf. A079583.
%K A157210 nonn,tabl
%O A157210 0,5
%A A157210 _Roger L. Bagula_, Feb 25 2009
%E A157210 Edited by _G. C. Greubel_, Jan 10 2022