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A155491 Triangle T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1), read by rows.

%I A155491 #8 Apr 03 2022 08:20:51
%S A155491 1,1,1,1,12,1,1,78,78,1,1,415,1820,415,1,1,2031,27410,27410,2031,1,1,
%T A155491 9534,330225,959350,330225,9534,1,1,43660,3488884,23935450,23935450,
%U A155491 3488884,43660,1,1,196569,33888576,484631574,1120179060,484631574,33888576,196569,1
%N A155491 Triangle T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1), read by rows.
%H A155491 G. C. Greubel, <a href="/A155491/b155491.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A155491 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 A155491 From _G. C. Greubel_, Apr 01 2022: (Start)
%F A155491 T(n, k) = binomial(n+1, k)*A142458(n+1, k+1)/(k+1).
%F A155491 T(n, n-k) = T(n, k). (End)
%e A155491 Triangle begins as:
%e A155491   1;
%e A155491   1,      1;
%e A155491   1,     12,        1;
%e A155491   1,     78,       78,         1;
%e A155491   1,    415,     1820,       415,          1;
%e A155491   1,   2031,    27410,     27410,       2031,         1;
%e A155491   1,   9534,   330225,    959350,     330225,      9534,        1;
%e A155491   1,  43660,  3488884,  23935450,   23935450,   3488884,    43660,      1;
%e A155491   1, 196569, 33888576, 484631574, 1120179060, 484631574, 33888576, 196569, 1;
%t A155491 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 A155491 T[n_, k_, m_]:= Binomial[n+1,k]*t[n+1,k+1,m]/(k+1);
%t A155491 Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Apr 01 2022 *)
%o A155491 (Sage)
%o A155491 @CachedFunction
%o A155491 def t(n,k,m):
%o A155491     if (k==1 or k==n): return 1
%o A155491     else: return (m*(n-k)+1)*t(n-1,k-1,m) + (m*k-m+1)*t(n-1,k,m)
%o A155491 def T(n,k,m): return binomial(n+1,k)*t(n+1,k+1,m)/(k+1)
%o A155491 flatten([[T(n,k,3) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Apr 01 2022
%Y A155491 Cf. A001263 (m=0), A155467 (m=1), this sequence (m=3), A155493 (m=4).
%Y A155491 Cf. A142458.
%K A155491 nonn,tabl
%O A155491 0,5
%A A155491 _Roger L. Bagula_, Jan 23 2009
%E A155491 Edited by _G. C. Greubel_, Apr 01 2022