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A135855
A007318 * a tridiagonal matrix with (1, 4, 1, 0, 0, 0, ...) in every column.
2
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
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
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Dec 01 2007
STATUS
approved