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%I #16 Sep 08 2022 08:45:49
%S 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,
%T 41,1,1,49,161,273,273,161,49,1,1,57,217,441,553,441,217,57,1,1,65,
%U 281,665,1001,1001,665,281,65,1,1,73,353,953,1673,2009,1673,953,353,73,1
%N Triangle T(n,k) = 8*binomial(n,k) - 7 with columns 0 <= k <= n.
%C Triangle T(n,k): the coefficient [x^k] of the polynomial 8*(x+1)^n -7*( x^(n+1) - 1)/(x-1).
%H G. C. Greubel, <a href="/A168625/b168625.txt">Rows n = 0..100 of triangle, flattened</a>
%F 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
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 9, 1;
%e 1, 17, 17, 1;
%e 1, 25, 41, 25, 1;
%e 1, 33, 73, 73, 33, 1;
%e 1, 41, 113, 153, 113, 41, 1;
%e 1, 49, 161, 273, 273, 161, 49, 1;
%e 1, 57, 217, 441, 553, 441, 217, 57, 1;
%e 1, 65, 281, 665, 1001, 1001, 665, 281, 65, 1;
%e 1, 73, 353, 953, 1673, 2009, 1673, 953, 353, 73, 1;
%p A168625:= (n,k) -> 8*binomial(n, k) -7; seq(seq(A168625(n, k), k = 0..n), n = 0.. 10); # _G. C. Greubel_, Mar 12 2020
%t m = 8; p[x_, n_]:= FullSimplify[ExpandAll[m*(x+1)^n -(m-1)(x^(n+1) -1)/(x-1)]];
%t Table[CoefficientList[p[x, n], x], {n,0,10}]//Flatten
%t Table[8*Binomial[n, k] -7, {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 12 2020 *)
%o (Magma) [8*Binomial(n, k) -7: k in [0..n], n in [0..10]]; // _G. C. Greubel_, Mar 12 2020
%o (Sage) [[8*binomial(n, k) -7 for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Mar 12 2020
%Y Sequence m*binomial(n,k) - (m-1): A007318 (m=1), A109128 (m=2), A131060 (m=3), A131061 (m=4), A131063 (m=5), A131065 (m=6), A131067 (m=7), this sequence (m=8).
%K nonn,easy,tabl
%O 0,5
%A _Roger L. Bagula_, Dec 01 2009
%E Definition simplified by _R. J. Mathar_, Sep 02 2011
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