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A135089
Triangle T(n,k) = 5*binomial(n,k) with T(0,0) = 1, read by rows.
2
1, 5, 5, 5, 10, 5, 5, 15, 15, 5, 5, 20, 30, 20, 5, 5, 25, 50, 50, 25, 5, 5, 30, 75, 100, 75, 30, 5, 5, 35, 105, 175, 175, 105, 35, 5, 5, 40, 140, 280, 350, 280, 140, 40, 5, 5, 45, 180, 420, 630, 630, 420, 180, 45, 5, 5, 50, 225, 600, 1050, 1260, 1050, 600, 225, 50, 5
OFFSET
0,2
COMMENTS
Row sums = A020714 (except for the first term).
Triangle T(n,k), 0 <= k <= n, read by rows given by (5, -4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (5, -4, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 24 2013
FORMULA
T(n,k) = 5*binomial(n,k), n > 0, 0 <= k <= n.
Equals 2*A134059(n,k) - A007318(n,k).
G.f.: (1+4*x+4*x*y)/(1-x-x*y). - Philippe Deléham, Nov 24 2013
Sum_{k=0..n} T(n,k) = A020714(n) - 4*[n=0]. - G. C. Greubel, May 03 2021
EXAMPLE
First few rows of the triangle:
1;
5, 5;
5, 10, 5;
5, 15, 15, 5;
5, 20, 30 20, 5;
5, 25, 50, 50, 25, 5;
5, 30, 75, 100, 75, 30, 5.
MATHEMATICA
Table[5*Binomial[n, k] -4*Boole[n==0], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Sep 22 2016; May 03 2021 *)
PROG
(Magma) [1] cat [5*Binomial(n, k): k in [0..n], n in [1..12]]; // G. C. Greubel, May 03 2021
(Sage)
def A135089(n, k): return 5*binomial(n, k) - 4*bool(n==0)
flatten([[A135089(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 03 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Nov 18 2007
STATUS
approved