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A168625
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Triangle T(n,k) = 8*binomial(n,k) - 7 with columns 0 <= k <= n.
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4
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1, 1, 1, 1, 9, 1, 1, 17, 17, 1, 1, 25, 41, 25, 1, 1, 33, 73, 73, 33, 1, 1, 41, 113, 153, 113, 41, 1, 1, 49, 161, 273, 273, 161, 49, 1, 1, 57, 217, 441, 553, 441, 217, 57, 1, 1, 65, 281, 665, 1001, 1001, 665, 281, 65, 1, 1, 73, 353, 953, 1673, 2009, 1673, 953, 353, 73, 1
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OFFSET
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0,5
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COMMENTS
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Triangle T(n,k): the coefficient [x^k] of the polynomial 8*(x+1)^n -7*( x^(n+1) - 1)/(x-1).
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LINKS
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FORMULA
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T(n,k) = [x^k] ( 8*(x+1)^n-7*Sum_{s=0..n} x^s ) = 8*A007318(n,k) - 7. - R. J. Mathar, Sep 02 2011
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 9, 1;
1, 17, 17, 1;
1, 25, 41, 25, 1;
1, 33, 73, 73, 33, 1;
1, 41, 113, 153, 113, 41, 1;
1, 49, 161, 273, 273, 161, 49, 1;
1, 57, 217, 441, 553, 441, 217, 57, 1;
1, 65, 281, 665, 1001, 1001, 665, 281, 65, 1;
1, 73, 353, 953, 1673, 2009, 1673, 953, 353, 73, 1;
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MAPLE
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MATHEMATICA
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m = 8; p[x_, n_]:= FullSimplify[ExpandAll[m*(x+1)^n -(m-1)(x^(n+1) -1)/(x-1)]];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
Table[8*Binomial[n, k] -7, {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 12 2020 *)
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PROG
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(Magma) [8*Binomial(n, k) -7: k in [0..n], n in [0..10]]; // G. C. Greubel, Mar 12 2020
(Sage) [[8*binomial(n, k) -7 for k in (0..n)] for n in (0..10)] # G. C. Greubel, Mar 12 2020
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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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