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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.
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%I #5 Jan 10 2022 18:21:24

%S 1,1,1,1,11,1,1,54,54,1,1,229,822,229,1,1,932,8368,8368,932,1,1,3747,

%T 72066,174758,72066,3747,1,1,15010,570006,2759750,2759750,570006,

%U 15010,1,1,60065,4297714,37366190,73850596,37366190,4297714,60065,1,1,240288,31495488,460448520,1591033788,1591033788,460448520,31495488,240288,1

%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="/A157209/b157209.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, 11, 1;

%e 1, 54, 54, 1;

%e 1, 229, 822, 229, 1;

%e 1, 932, 8368, 8368, 932, 1;

%e 1, 3747, 72066, 174758, 72066, 3747, 1;

%e 1, 15010, 570006, 2759750, 2759750, 570006, 15010, 1;

%e 1, 60065, 4297714, 37366190, 73850596, 37366190, 4297714, 60065, 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,0,10}, {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): # A157209

%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..20)]) # _G. C. Greubel_, Jan 10 2022

%Y Cf. A007318 (m=0), A157207 (m=1), A157208 (m=2), this sequence (m=3).

%Y Cf. A157147, A157148, A157149, A157150, A157151, A157152, A157153, A157154, A157155, A157156, A157210, A157211, A157212, 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