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%I #9 Nov 10 2013 06:23:52
%S 1,1,2,2,5,8,3,8,14,20,4,11,20,30,40,5,14,26,40,55,70,6,17,32,50,70,
%T 91,112,7,20,38,60,85,112,140,168,8,23,44,70,100,133,168,204,240,9,26,
%U 50,80,115,154,196,240,285,330,10,29,56,90,130,175,224,276,330
%N Triangular array: the self-fusion of (p(n,x)), where p(n,x)=x^n+2x^(n-1)+3x^(n-2)+...+nx+(n+1).
%C See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
%H Reinhard Zumkeller, <a href="/A193891/b193891.txt">Table of n, a(n) for n = 0..8000</a>
%e First six rows:
%e 1
%e 1...2
%e 2...5....8
%e 3...8....14...20
%e 4...11...20...30...40
%e 5...14...26...40...55...70
%t z = 9;
%t p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + n + 1;
%t q[n_, x_] := p[n, x];
%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t g[n_] := CoefficientList[w[n, x], {x}]
%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193891 *)
%t TableForm[Table[g[n], {n, -1, z}]]
%t Flatten[Table[g[n], {n, -1, z}]] (* A193892 *)
%o (Haskell)
%o a193891 n k = a193891_tabl !! n !! k
%o a193891_row n = a193891_tabl !! n
%o a193891_tabl = [1] : map fst (iterate
%o (\(xs, i) -> (zipWith (+) (0:xs) [i, 2 * i ..], i + 1)) ([1,2], 2))
%o -- _Reinhard Zumkeller_, Nov 10 2013
%Y Cf. A193722, A193892.
%K nonn,tabl
%O 0,3
%A _Clark Kimberling_, Aug 08 2011
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