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Triangle T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1), read by rows.
4

%I #8 Apr 03 2022 08:20:51

%S 1,1,1,1,12,1,1,78,78,1,1,415,1820,415,1,1,2031,27410,27410,2031,1,1,

%T 9534,330225,959350,330225,9534,1,1,43660,3488884,23935450,23935450,

%U 3488884,43660,1,1,196569,33888576,484631574,1120179060,484631574,33888576,196569,1

%N Triangle T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1), read by rows.

%H G. C. Greubel, <a href="/A155491/b155491.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = binomial(n+1, k)*t(n, k, m)/(k+1), where t(n,k,m) = (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m), t(n,1,m) = t(n,n,m) = 1, and m = 3.

%F From _G. C. Greubel_, Apr 01 2022: (Start)

%F T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1).

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

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 12, 1;

%e 1, 78, 78, 1;

%e 1, 415, 1820, 415, 1;

%e 1, 2031, 27410, 27410, 2031, 1;

%e 1, 9534, 330225, 959350, 330225, 9534, 1;

%e 1, 43660, 3488884, 23935450, 23935450, 3488884, 43660, 1;

%e 1, 196569, 33888576, 484631574, 1120179060, 484631574, 33888576, 196569, 1;

%t t[n_, k_, m_]:= t[n,k,m]= If[k==1 || k==n, 1, (m*n-m*k+1)*t[n-1,k-1,m] + (m*k -(m -1))*t[n-1,k,m]];

%t T[n_, k_, m_]:= Binomial[n+1,k]*t[n+1,k+1,m]/(k+1);

%t Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 01 2022 *)

%o (Sage)

%o @CachedFunction

%o def t(n,k,m):

%o if (k==1 or k==n): return 1

%o else: return (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m)

%o def T(n,k,m): return binomial(n+1,k)*t(n+1,k+1,m)/(k+1)

%o flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 01 2022

%Y Cf. A001263 (m=0), A155467 (m=1), this sequence (m=3), A155493 (m=4).

%Y Cf. A142458.

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_, Jan 23 2009

%E Edited by _G. C. Greubel_, Apr 01 2022