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A131113
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T(n,k) = 5*binomial(n,k) - 4*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, 5, 1, 5, 10, 1, 5, 15, 15, 1, 5, 20, 30, 20, 1, 5, 25, 50, 50, 25, 1, 5, 30, 75, 100, 75, 30, 1, 5, 35, 105, 175, 175, 105, 35, 1, 5, 40, 140, 280, 350, 280, 140, 40, 1, 5, 45, 180, 420, 630, 630, 420, 180, 45, 1
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OFFSET
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0,2
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COMMENTS
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Row sums = A048487: (1, 6, 16, 36, 76, 156, ...).
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LINKS
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FORMULA
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T(n,k) = 5*A007318(n,k) - 4*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.
Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 4*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;
5, 1;
5, 10, 1;
5, 15, 15, 1;
5, 20, 30, 20, 1;
5, 25, 50, 50, 25, 1;
5, 30, 75, 100, 75, 30, 1;
...
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MAPLE
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seq(seq(`if`(k=n, 1, 5*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, 5*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, 5*binomial(n, k)); \\ G. C. Greubel, Nov 18 2019
(Magma) [k eq n select 1 else 5*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 5*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 5*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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