cp's OEIS Frontend

A157212 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 = 3, read by rows.

%I A157212 #6 Jan 11 2022 02:19:06
%S A157212 1,1,1,1,5,1,1,24,24,1,1,103,306,103,1,1,422,3028,3028,422,1,1,1701,
%T A157212 26064,57806,26064,1701,1,1,6820,207132,889640,889640,207132,6820,1,1,
%U A157212 27299,1569298,11975936,22436968,11975936,1569298,27299,1,1,109218,11544744,147711834,472619880,472619880,147711834,11544744,109218,1
%N A157212 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 = 3, read by rows.
%H A157212 G. C. Greubel, <a href="/A157212/b157212.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157212 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 = 3.
%F A157212 T(n, n-k, m) = T(n, k, m).
%e A157212 Triangle begins as:
%e A157212   1;
%e A157212   1,     1;
%e A157212   1,     5,       1;
%e A157212   1,    24,      24,        1;
%e A157212   1,   103,     306,      103,        1;
%e A157212   1,   422,    3028,     3028,      422,        1;
%e A157212   1,  1701,   26064,    57806,    26064,     1701,       1;
%e A157212   1,  6820,  207132,   889640,   889640,   207132,    6820,     1;
%e A157212   1, 27299, 1569298, 11975936, 22436968, 11975936, 1569298, 27299, 1;
%t A157212 f[n_,k_]:= If[k<=Floor[n/2], k, n-k];
%t A157212 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 A157212 Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 10 2022 *)
%o A157212 (Sage)
%o A157212 def f(n,k): return k if (k <= n//2) else n-k
%o A157212 @CachedFunction
%o A157212 def T(n,k,m):  # A157212
%o A157212     if (k==0 or k==n): return 1
%o A157212     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 A157212 flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 10 2022
%Y A157212 Cf. A007318 (m=0), A157210 (m=1), A157211 (m=2), this sequence (m=3).
%Y A157212 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157268, A157272, A157273, A157274, A157275.
%K A157212 nonn,tabl
%O A157212 0,5
%A A157212 _Roger L. Bagula_, Feb 25 2009
%E A157212 Edited by _G. C. Greubel_, Jan 10 2022