cp's OEIS Frontend

A157278 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 = 3, read by rows.

%I A157278 #6 Feb 06 2022 04:02:42
%S A157278 1,1,1,1,14,1,1,69,69,1,1,292,1134,292,1,1,1187,11686,11686,1187,1,1,
%T A157278 4770,100737,254132,100737,4770,1,1,19105,795723,4061249,4061249,
%U A157278 795723,19105,1,1,76448,5990296,55157324,111691642,55157324,5990296,76448,1
%N A157278 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 = 3, read by rows.
%H A157278 G. C. Greubel, <a href="/A157278/b157278.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A157278 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 = 3.
%F A157278 T(n, n-k, m) = T(n, k, m).
%e A157278 Triangle begins as:
%e A157278   1;
%e A157278   1,     1;
%e A157278   1,    14,       1;
%e A157278   1,    69,      69,        1;
%e A157278   1,   292,    1134,      292,         1;
%e A157278   1,  1187,   11686,    11686,      1187,        1;
%e A157278   1,  4770,  100737,   254132,    100737,     4770,       1;
%e A157278   1, 19105,  795723,  4061249,   4061249,   795723,   19105,     1;
%e A157278   1, 76448, 5990296, 55157324, 111691642, 55157324, 5990296, 76448, 1;
%t A157278 f[n_,k_]:= If[k<=Floor[n/2], 2*k, 2*(n-k)];
%t A157278 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 A157278 Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 06 2022 *)
%o A157278 (Sage)
%o A157278 def f(n,k): return 2*k if (k <= n//2) else 2*(n-k)
%o A157278 @CachedFunction
%o A157278 def T(n,k,m):  # A157278
%o A157278     if (k==0 or k==n): return 1
%o A157278     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 A157278 flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 06 2022
%Y A157278 Cf. A007318 (m=0), A157275 (m=1), A157277 (m=2), this sequence (m=3).
%Y A157278 Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274.
%K A157278 nonn,tabl
%O A157278 0,5
%A A157278 _Roger L. Bagula_, Feb 26 2009
%E A157278 Edited by _G. C. Greubel_, Feb 06 2022