A133138 - OEIS

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

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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: A338898 A376115 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