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A131112
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T(n,k) = 4*binomial(n,k) - 3*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).
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6
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1, 4, 1, 4, 8, 1, 4, 12, 12, 1, 4, 16, 24, 16, 1, 4, 20, 40, 40, 20, 1, 4, 24, 60, 80, 60, 24, 1, 4, 28, 84, 140, 140, 84, 28, 1, 4, 32, 112, 224, 280, 224, 112, 32, 1, 4, 36, 144, 336, 504, 504, 336, 144, 36, 1
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OFFSET
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0,2
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LINKS
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FORMULA
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T(n,k) = 4*A007318(n,k) - 3*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.
n-th row sum = A036563(n+2) = 2^(n+2) - 3.
Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 3*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021
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EXAMPLE
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Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
1;
4, 1;
4, 8, 1;
4, 12, 12, 1;
4, 16, 24, 16, 1;
4, 20, 40, 40, 20, 1;
...
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MAPLE
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seq(seq(`if`(k=n, 1, 4*binomial(n, k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019
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MATHEMATICA
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Table[If[k==n, 1, 4*Binomial[n, k]], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)
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PROG
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(PARI) T(n, k) = if(k==n, 1, 4*binomial(n, k)); \\ G. C. Greubel, Nov 18 2019
(Magma) [k eq n select 1 else 4*Binomial(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019
(Sage)
def T(n, k):
if (k==n): return 1
else: return 4*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)]
(GAP)
T:= function(n, k)
if k=n then return 1;
else return 4*Binomial(n, k);
fi; end;
Flat(List([0..10], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Nov 18 2019
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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