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A135855 A007318 * a tridiagonal matrix with (1, 4, 1, 0, 0, 0, ...) in every column.

%I A135855 #9 Feb 07 2022 02:22:51
%S A135855 1,5,1,10,6,1,16,16,7,1,23,32,23,8,1,31,55,55,31,9,1,40,86,110,86,40,
%T A135855 10,1,50,126,196,196,126,50,11,1,61,176,322,392,322,176,61,12,1,73,
%U A135855 237,498,714,714,498,237,73,13,1
%N A135855 A007318 * a tridiagonal matrix with (1, 4, 1, 0, 0, 0, ...) in every column.
%H A135855 G. C. Greubel, <a href="/A135855/b135855.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A135855 Binomial transform of an infinite tridiagonal matrix with (1, 4, 1, 0, 0, 0, ...) in every column; i.e., (1, 1, 1, ...) in the main diagonal, (4, 4, 4, 0, 0, 0, ...) in the subdiagonal and (1, 1, 1, ...) in the subsubdiagonal.
%F A135855 T(n, 0) = A052905(n).
%F A135855 Sum_{k=0..n} T(n, k) = A101945(n).
%F A135855 From _G. C. Greubel_, Feb 06 2022: (Start)
%F A135855 T(n, k) = T(n-1, k-1) + T(n-1, k), with T(n, n) = 1, T(n, 0) = A052905(n).
%F A135855 T(n, k) = binomial(n,k)*(n^2 + (2*k+7)*n - 2*(k^2 + 2*k -1))/((k+1)*(k+2)).
%F A135855 T(n, 1) = A134465(n).
%F A135855 T(n, 2) = A022815(n-1).
%F A135855 T(n, n-1) = n+3.
%F A135855 T(n, n-2) = A052905(n+2). (End)
%e A135855 First few rows of the triangle:
%e A135855    1;
%e A135855    5,  1;
%e A135855   10,  6,   1;
%e A135855   16, 16,   7,  1;
%e A135855   23, 32,  23,  8,  1;
%e A135855   31, 55,  55, 31,  9,  1;
%e A135855   40, 86, 110, 86, 40, 10, 1;
%e A135855   ...
%t A135855 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, (n^2+7*n+2)/2, If[k==n, 1, T[n-1, k-1] + T[n-1, k]]]];
%t A135855 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 06 2022 *)
%o A135855 (Magma)
%o A135855 A135855:= func< n,k | Binomial(n,k)*(n^2 + (2*k+7)*n - 2*(k^2 + 2*k -1))/((k+1)*(k+2)) >;
%o A135855 [A135855(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Feb 06 2022
%o A135855 (Sage)
%o A135855 @CachedFunction
%o A135855 def T(n,k): # A135855
%o A135855     if (k==0): return (n^2+7*n+2)/2
%o A135855     elif (k==n): return 1
%o A135855     else: return T(n-1, k-1) + T(n-1, k)
%o A135855 flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # _G. C. Greubel_, Feb 06 2022
%Y A135855 Cf. A007318, A022815, A052905, A101945, A134465.
%K A135855 nonn,tabl
%O A135855 0,2
%A A135855 _Gary W. Adamson_, Dec 01 2007