A158208 Triangle read by rows: p(x,n) = 2 if n = 0, Sum_{i=0..floor((n-1)/2)} binomial(n, i)*(x - 1)^i + x^n*Sum_{i=0..floor((n-1)/2)} binomial(n, i)*(1/x - 1)^i otherwise.

2, 1, 1, 1, 0, 1, -2, 3, 3, -2, -3, 4, 0, 4, -3, 6, -15, 10, 10, -15, 6, 10, -24, 15, 0, 15, -24, 10, -20, 70, -84, 35, 35, -84, 70, -20, -35, 120, -140, 56, 0, 56, -140, 120, -35, 70, -315, 540, -420, 126, 126, -420, 540, -315, 70, 126, -560, 945, -720, 210, 0, 210

Offset

0,1

Comments

The first half of every second row gives the coefficients of a polynomial approximation of f(0) = f'(0) = f'(1) = f''(0) = f''(1) = ... = 0 and f(1)=1: x, -2x^3 + 3x^2, 6x^5 - 15x^4 + 10x^3, ... - Martin Clever, Sep 12 2022

Links

Table of n, a(n) for n=0..61.

Example

Triangle begins:
    2;
    1,    1;
    1,    0,    1;
   -2,    3,    3,   -2;
   -3,    4,    0,    4,  -3;
    6,  -15,   10,   10, -15,   6;
   10,  -24,   15,    0,  15, -24,   10;
  -20,   70,  -84,   35,  35, -84,   70,  -20;
  -35,  120, -140,   56,   0,  56, -140,  120,  -35;
   70, -315,  540, -420, 126, 126, -420,  540, -315,   70;
  126, -560,  945, -720, 210,   0,  210, -720,  945, -560, 126;
  ...

Mathematica

p[x_, n_] = If[ n == 0, 2, Sum[Binomial[ n, i]*(x - 1)^i, {i, 0, Floor[(n - 1)/2]}] + Expand[x^n*Sum[Binomial[n, i]*(1/x - 1)^ i, {i, 0, Floor[(n - 1)/2]}]]];
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]

Crossrefs

Sequence in context: A035212 A318133 A068029 * A348652 A117274 A221650

Adjacent sequences: A158205 A158206 A158207 * A158209 A158210 A158211

Keyword

sign,tabl,uned,less

Author

Roger L. Bagula, Mar 13 2009

Status

approved