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A135227
Triangle A000012 * A135225, read by rows.
2
1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 6, 4, 1, 6, 5, 10, 10, 5, 1, 7, 6, 15, 20, 15, 6, 1, 8, 7, 21, 35, 35, 21, 7, 1, 9, 8, 28, 56, 70, 56, 28, 8, 1, 10, 9, 36, 84, 126, 126, 84, 36, 9, 1, 11, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
OFFSET
0,2
COMMENTS
Row sums = A006127: (1, 3, 6, 11, 20, 37, ...).
FORMULA
A000012 * A135225 as infinite lower triangular matrices. Left border of 1's in Pascal's Triangle (A007318) is replaced with a column of (1,2,3,...).
T(n,k) = binomial(n,k), with T(n,0) = n+1. - G. C. Greubel, Nov 20 2019
EXAMPLE
First few rows of the triangle:
1;
2, 1;
3, 2, 1;
4, 3, 3, 1;
5, 4, 6, 4, 1;
6, 5, 10, 10, 5, 1;
7, 6, 15, 20, 15, 6, 1;
...
MAPLE
seq(seq( `if`(k=0, n+1, binomial(n, k)), k=0..n), n=0..12); # G. C. Greubel, Nov 20 2019
MATHEMATICA
Table[If[k==0, n+1, Binomial[n, k]], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)
PROG
(PARI) T(n, k) = if(k==0, n+1, binomial(n, k)); \\ G. C. Greubel, Nov 20 2019
(Magma) [k eq 0 select n+1 else Binomial(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2019
(Sage)
def T(n, k):
if (k==0): return 1
else: return binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 20 2019
(GAP)
T:= function(n, k)
if k=0 then return 1;
else return Binomial(n, k);
fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 20 2019
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Nov 23 2007
EXTENSIONS
More terms added by G. C. Greubel, Nov 20 2019
STATUS
approved