%I #21 May 19 2018 02:29:27

%S 2,2,2,1,3,2,1,3,4,2,1,4,6,5,2,1,5,10,10,6,2,1,6,15,20,15,7,2,1,7,21,

%T 35,35,21,8,2,1,8,28,56,70,56,28,9,2,1,9,36,84,126,126,84,36,10,2,1,

%U 10,45,120,210,252,210,120,45,11,2

%N Triangle T(n,k) of the coefficients of the polynomials Q(n,x)=(1+x)[(1+x)^(n-1)+x^(n-1)], Q(0,x)=2.

%F From _R. J. Mathar_, Jun 12 2008: (Start)

%F T(n,k) = A007318(n,k), 0 <= k < n-1.

%F T(n,k) = A007318(n,k)+1, n-1 <= k <= n.

%F Sum_{k=0..n} T(n,k) = A133140(n). (End)

%F T(n,k) = A007318(n,k) + A097806(n,k). - _Franck Maminirina Ramaharo_, May 18 2018

%e Triangle T(n,k) begins:

%e n/k 0 1 2 3 4 5 6 7 8 9 10 11 12

%e 0: 2

%e 1: 2 2

%e 2: 1 3 2

%e 3: 1 3 4 2

%e 4: 1 4 6 5 2

%e 5: 1 5 10 10 6 2

%e 6: 1 6 15 20 15 7 2

%e 7: 1 7 21 35 35 21 8 2

%e 8: 1 8 28 56 70 56 28 9 2

%e 9: 1 9 36 84 126 126 84 36 10 2

%e 10: 1 10 45 120 210 252 210 120 45 11 2

%e 11: 1 11 55 165 330 462 462 330 165 55 12 2

%e 12: 1 12 66 220 495 792 924 792 495 220 66 13 2

%e ... - _Franck Maminirina Ramaharo_, May 18 2018

%t q[n_] := (1+x)*((1+x)^(n-1) + x^(n-1)); t[n_, k_] := Coefficient[q[n], x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Dec 16 2013 *)

%o (Maxima)

%o Q(n, x) := (1 + x)*((1 + x)^(n - 1) + x^(n - 1))$

%o t(n,k) := ratcoef(expand(Q(n, x)), x, k)$

%o for n:0 thru 20 do print(makelist(t(n, k), k, 0, n)); /* _Franck Maminirina Ramaharo_, May 18 2018 */

%Y Cf. A133135.

%K nonn,tabl

%O 0,1

%A _Paul Curtz_, Sep 21 2007

%E Edited by _R. J. Mathar_, Jun 12 2008