A133138 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.

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

Offset

0,1

Links

Table of n, a(n) for n=0..65.

Formula

From R. J. Mathar, Jun 12 2008: (Start)

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

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

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

T(n,k) = A007318(n,k) + A097806(n,k). - Franck Maminirina Ramaharo, May 18 2018

Example

Triangle T(n,k) begins:
n/k 0   1   2    3    4    5    6    7    8    9  10  11  12
0:  2
1:  2   2
2:  1   3   2
3:  1   3   4    2
4:  1   4   6    5    2
5:  1   5  10   10    6    2
6:  1   6  15   20   15    7    2
7:  1   7  21   35   35   21    8    2
8:  1   8  28   56   70   56   28    9    2
9:  1   9  36   84  126  126   84   36   10    2
10: 1  10  45  120  210  252  210  120   45   11   2
11: 1  11  55  165  330  462  462  330  165   55  12   2
12: 1  12  66  220  495  792  924  792  495  220  66  13   2
... - Franck Maminirina Ramaharo, May 18 2018

Mathematica

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 *)

Prog

(Maxima)
Q(n, x) := (1 + x)*((1 + x)^(n - 1) + x^(n - 1))$
t(n, k) := ratcoef(expand(Q(n, x)), x, k)$
for n:0 thru 20 do print(makelist(t(n, k), k, 0, n)); /* Franck Maminirina Ramaharo, May 18 2018 */

Crossrefs

Cf. A133135.

Sequence in context: A230243 A338898 A078687 * A348193 A194326 A295277

Adjacent sequences: A133135 A133136 A133137 * A133139 A133140 A133141

Keyword

nonn,tabl

Author

Paul Curtz, Sep 21 2007

Extensions

Edited by R. J. Mathar, Jun 12 2008

Status

approved