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A124928
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Triangle read by rows: T(n,0) = 1, T(n,k) = 3*binomial(n,k) if k>=0 (0<=k<=n).
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3
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1, 1, 3, 1, 6, 3, 1, 9, 9, 3, 1, 12, 18, 12, 3, 1, 15, 30, 30, 15, 3, 1, 18, 45, 60, 45, 18, 3, 1, 21, 63, 105, 105, 63, 21, 3, 1, 24, 84, 168, 210, 168, 84, 24, 3, 1, 27, 108, 252, 378, 378, 252, 108, 27, 3, 1, 30, 135, 360, 630, 756, 630, 360, 135, 30, 3
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OFFSET
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0,3
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COMMENTS
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Row sums = A033484: (1, 4, 10, 22, 46, 94...); 3*2^n - 2.
Analogous triangle using (1,2,2,2...) as the main diagonal of M = A124927.
Except for the first column, entries in the Pascal triangle are tripled.
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LINKS
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FORMULA
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G.f.: G(t,z) = 3/(1-(1+t)*z) - 2/(1-z).
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EXAMPLE
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First few rows of the triangle are:
1;
1, 3;
1, 6, 3;
1, 9, 9, 3;
1, 12, 18, 12, 3;
1, 15, 30, 30, 15, 3;
1, 18, 45, 60, 45, 18, 3;
...
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MAPLE
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T:=proc(n, k) if k=0 then 1 else 3*binomial(n, k) fi end: for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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MATHEMATICA
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Flatten[Table[If[k==0, 1, 3*Binomial[n, k]], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Oct 19 2013 *)
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PROG
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(PARI) T(n, k) = if(k==0, 1, 3*binomial(n, k)); \\ G. C. Greubel, Nov 19 2019
(Magma) [k eq 0 select 1 else 3*Binomial(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 19 2019
(Sage)
def T(n, k):
if (k==0): return 1
else: return 3*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 19 2019
(GAP)
T:= function(n, k)
if k=0 then return 1;
else return 3*Binomial(n, k);
fi; end;
Flat(List([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 19 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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