A157212 - OEIS

%I #6 Jan 11 2022 02:19:06

%S 1,1,1,1,5,1,1,24,24,1,1,103,306,103,1,1,422,3028,3028,422,1,1,1701,

%T 26064,57806,26064,1701,1,1,6820,207132,889640,889640,207132,6820,1,1,

%U 27299,1569298,11975936,22436968,11975936,1569298,27299,1,1,109218,11544744,147711834,472619880,472619880,147711834,11544744,109218,1

%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) = k if k <= floor(n/2) otherwise n-k, and m = 3, read by rows.

%H G. C. Greubel, <a href="/A157212/b157212.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) = k if k <= floor(n/2) otherwise 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,     5,       1;

%e   1,    24,      24,        1;

%e   1,   103,     306,      103,        1;

%e   1,   422,    3028,     3028,      422,        1;

%e   1,  1701,   26064,    57806,    26064,     1701,       1;

%e   1,  6820,  207132,   889640,   889640,   207132,    6820,     1;

%e   1, 27299, 1569298, 11975936, 22436968, 11975936, 1569298, 27299, 1;

%t f[n_,k_]:= If[k<=Floor[n/2], k, 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_, Jan 10 2022 *)

%o (Sage)

%o def f(n,k): return k if (k <= n//2) else n-k

%o @CachedFunction

%o def T(n,k,m):  # A157212

%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_, Jan 10 2022

%Y Cf. A007318 (m=0), A157210 (m=1), A157211 (m=2), this sequence (m=3).

%Y Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157207, A157208, A157209, A157268, A157272, A157273, A157274, A157275.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Feb 25 2009

%E Edited by _G. C. Greubel_, Jan 10 2022