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Triangle read by rows: row n is the expansion of p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[(2^(m - 1) +n*m - n + 1)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}].
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%I #8 Oct 04 2024 08:53:29

%S 1,1,1,1,4,1,1,11,11,1,1,34,34,34,1,1,109,102,102,109,1,1,350,303,292,

%T 303,350,1,1,1127,901,819,819,901,1127,1,1,3688,2716,2296,2182,2296,

%U 2716,3688,1,1,12425,8420,6548,5822,5822,6548,8420,12425,1,1,43402

%N Triangle read by rows: row n is the expansion of p(x,n)=If[n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[(2^(m - 1) +n*m - n + 1)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}].

%F T(n,0) = T(n,n) = 1. For 0 < m < n, T(n,m) = binomial(n,m) + 2^(n-3)*(2^(m-1)+2^(n-m-1)+n^2-2*n+2). - _Jason Yuen_, Oct 03 2024

%e Triangle begins:

%e {1}

%e {1, 1}

%e {1, 4, 1}

%e {1, 11, 11, 1}

%e {1, 34, 34, 34, 1}

%e {1, 109, 102, 102, 109, 1}

%e {1, 350, 303, 292, 303, 350, 1}

%e {1, 1127, 901, 819, 819, 901, 1127, 1}

%e {1, 3688, 2716, 2296, 2182, 2296, 2716, 3688, 1}

%e {1, 12425, 8420, 6548, 5822, 5822, 6548, 8420, 12425, 1}

%e {1, 43402, 27181, 19320, 15826, 14844, 15826, 19320, 27181, 43402, 1}

%t p[x_, n_] = If[n == 0, 1, (x + 1)^n + 2^(n - 3)*Sum[(2^(m - 1) + n*m - n + 1)*x^m*(1 + x^(n - 2*m)), {m, 1, n - 1}]];

%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

%t Flatten[%]

%o (PARI) T(n,m) = if(m==0 || m==n, 1, binomial(n,m) + 2^(n-3)*(2^(m-1)+2^(n-m-1)+n^2-2*n+2)) \\ _Jason Yuen_, Oct 03 2024

%K nonn,tabl,easy,less

%O 0,5

%A _Roger L. Bagula_, Nov 03 2008