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A symmetrical triangle of coefficients based on A000217: a(n) = Binomial[n + 2, 2]; t(n,m)=a(n - m + 1)*a(m + 1).
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%I #3 Oct 12 2012 14:54:49

%S 9,18,18,30,36,30,45,60,60,45,63,90,100,90,63,84,126,150,150,126,84,

%T 108,168,210,225,210,168,108,135,216,280,315,315,280,216,135,165,270,

%U 360,420,441,420,360,270,165,198,330,450,540,588,588,540,450,330,198,234

%N A symmetrical triangle of coefficients based on A000217: a(n) = Binomial[n + 2, 2]; t(n,m)=a(n - m + 1)*a(m + 1).

%C Row sums are:

%C {9, 36, 96, 210, 406, 720, 1197, 1892, 2871, 4212, 6006}.

%D G. E. Andrews, Number Theory, 1971, Dover Publications New York, p 44.

%F a(n) = Binomial[n + 2, 2]; t(n,m)=a(n - m + 1)*a(m + 1).

%e {9},

%e {18, 18},

%e {30, 36, 30},

%e {45, 60, 60, 45},

%e {63, 90, 100, 90, 63},

%e {84, 126, 150, 150, 126, 84},

%e {108, 168, 210, 225, 210, 168, 108},

%e {135, 216, 280, 315, 315, 280, 216, 135},

%e {165, 270, 360, 420, 441, 420, 360, 270, 165},

%e {198, 330, 450, 540, 588, 588, 540, 450, 330, 198},

%e {234, 396, 550, 675, 756, 784, 756, 675, 550, 396, 234}

%t Clear[a, n, m, t] (*A000217*) a[0] = 1; a[1] = 3; a[n_] := a[n] = Binomial[n + 2, 2]; Table[a[n], {n, 0, 30}]; t[n_, m_] = a[n - m + 1]*a[m + 1]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]

%Y Cf. A000217.

%K nonn,tabl

%O 1,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Aug 25 2008