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

0,2

Links

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

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)

Example

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;
  ...

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 *)

Prog

(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
(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

Crossrefs

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

Sequence in context: A348689 A329752 A363969 * A116547 A013612 A112830

Adjacent sequences: A135852 A135853 A135854 * A135856 A135857 A135858

Keyword

nonn,tabl

Author

Gary W. Adamson, Dec 01 2007

Status

approved