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%I #10 Jan 13 2019 12:07:09
%S 1,1,1,1,1,1,1,1,2,2,2,1,1,2,3,4,3,2,1,1,3,5,7,7,5,3,1,1,3,6,10,12,12,
%T 10,6,3,1,1,4,9,16,22,24,22,16,9,4,1,1,4,10,20,31,40,44,40,31,20,10,4,
%U 1,1,5,14,30,51,71,84,84,71,51,30,14,5,1,1,5,15,35,65,101,135,155,155,135,101,65,35,15,5,1
%N Triangle read by rows (n >= 1): T(n,k) = [x^k] p(x,n), where p(x,n) = (x^3 + x^2 + x + 1)^floor(n/2) if n is odd, and p(x,n) = (x + 1)*p(x,n-1) otherwise.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 1, 1, 1;
%e 1, 2, 2, 2, 1;
%e 1, 2, 3, 4, 3, 2, 1;
%e 1, 3, 5, 7, 7, 5, 3, 1;
%e 1, 3, 6, 10, 12, 12, 10, 6, 3, 1;
%e 1, 4, 9, 16, 22, 24, 22, 16, 9, 4, 1;
%e 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1;
%e 1, 5, 14, 30, 51, 71, 84, 84, 71, 51, 30, 14, 5, 1;
%e 1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1;
%e 1, 6, 20, 50, 100, 166, 236, 290, 310, 290, 236, 166, 100, 50, 20, 6, 1;
%e ...
%t p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^3 + x^2 + x + 1)^Floor[n/2]]
%t Flatten[Table[CoefficientList[p[x, n], x], {n, 1, 12}]]
%o (Maxima)
%o p(x, n) := if mod(n, 2) = 0 then (x + 1)*p(x, n - 1) else (x^3 + x^2 + x + 1)^floor(n/2)$
%o T(n, k) := ratcoef(p(x, n), x, k)$
%o create_list(T(n, k), n, 1, 12, k, 0, hipow(fullratsimp(p(x, n)), x));
%o /* _Franck Maminirina Ramaharo_, Jan 13 2019 */
%Y Row sums: A000079.
%Y Cf. A008406, A171414.
%K nonn,easy,tabf
%O 1,9
%A _Roger L. Bagula_, Dec 08 2009
%E Edited by _Franck Maminirina Ramaharo_, Jan 13 2019
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