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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified December 3 00:23 EST 2022. Contains 358510 sequences. (Running on oeis4.)