A157278 - OEIS

%I #6 Feb 06 2022 04:02:42

%S 1,1,1,1,14,1,1,69,69,1,1,292,1134,292,1,1,1187,11686,11686,1187,1,1,

%T 4770,100737,254132,100737,4770,1,1,19105,795723,4061249,4061249,

%U 795723,19105,1,1,76448,5990296,55157324,111691642,55157324,5990296,76448,1

%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 = 3, read by rows.

%H G. C. Greubel, <a href="/A157278/b157278.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 = 3.

%F T(n, n-k, m) = T(n, k, m).

%e Triangle begins as:

%e   1;

%e   1,     1;

%e   1,    14,       1;

%e   1,    69,      69,        1;

%e   1,   292,    1134,      292,         1;

%e   1,  1187,   11686,    11686,      1187,        1;

%e   1,  4770,  100737,   254132,    100737,     4770,       1;

%e   1, 19105,  795723,  4061249,   4061249,   795723,   19105,     1;

%e   1, 76448, 5990296, 55157324, 111691642, 55157324, 5990296, 76448, 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,3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Feb 06 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):  # A157278

%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,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 06 2022

%Y Cf. A007318 (m=0), A157275 (m=1), A157277 (m=2), this sequence (m=3).

%Y Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157210, A157211, A157212, A157268, A157272, A157273, A157274.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 26 2009

%E Edited by _G. C. Greubel_, Feb 06 2022