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

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, 10, 1, 50, 126, 196, 196, 126, 50, 11, 1, 61, 176, 322, 392, 322, 176, 61, 12, 1, 73, 237, 498, 714, 714, 498, 237, 73, 13, 1

Offset: 0

Author

Gary W. Adamson, Dec 01 2007

Keywords

• nonn
• tabl

Examples

			First few rows of the triangle:
   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, 10, 1;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A007318, A022815, A052905, A101945, A134465.

Programs

Magma

A135855:= func< n,k | Binomial(n,k)*(n^2 + (2*k+7)*n - 2*(k^2 + 2*k -1))/((k+1)*(k+2)) >;
[A135855(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 06 2022

Mathematica

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]]]];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 06 2022 *)

Sage

@CachedFunction
def T(n,k): # A135855
    if (k==0): return (n^2+7*n+2)/2
    elif (k==n): return 1
    else: return T(n-1, k-1) + T(n-1, k)
flatten([[T(n,k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Feb 06 2022

Formula

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.

T(n, 0) = A052905(n).

Sum_{k=0..n} T(n, k) = A101945(n).

From G. C. Greubel, Feb 06 2022: (Start)

T(n, k) = T(n-1, k-1) + T(n-1, k), with T(n, n) = 1, T(n, 0) = A052905(n).

T(n, k) = binomial(n,k)*(n^2 + (2*k+7)*n - 2*(k^2 + 2*k -1))/((k+1)*(k+2)).

T(n, 1) = A134465(n).

T(n, 2) = A022815(n-1).

T(n, n-1) = n+3.

T(n, n-2) = A052905(n+2). (End)