A303700 Triangle read by rows in which row n gives coefficients of polynomial f_n(x)/(n+1) of degree less than n that satisfies Integral_{x=0..1} g(t - x) * f_n(x) dx = g(t) for any polynomial g(x) of degree less than n.

1, 2, -3, 3, -12, 10, 4, -30, 60, -35, 5, -60, 210, -280, 126, 6, -105, 560, -1260, 1260, -462, 7, -168, 1260, -4200, 6930, -5544, 1716, 8, -252, 2520, -11550, 27720, -36036, 24024, -6435, 9, -360, 4620, -27720, 90090, -168168, 180180, -102960, 24310

Offset

0,2

Links

Seiichi Manyama, Rows n = 0..139, flattened

Formula

f_n(x)/(n+1) = 1/(n!*x) * d^n/dx^n x^{n+1}*(1-x)^n.

T(n,k) = (-1)^(k)*(n+k+1)!*(k+1)/((k+1)!^2*(n-k)!). - Jacob Fauman, Sep 20 2022

Example

Triangle begins:
n | 0     1     2      3     4      5     6
--*-----------------------------------------
0 | 1;
1 | 2,   -3;
2 | 3,  -12,   10;
3 | 4,  -30,   60,   -35;
4 | 5,  -60,  210,  -280,  126;
5 | 6, -105,  560, -1260, 1260,  -462;
6 | 7, -168, 1260, -4200, 6930, -5544, 1716;

Crossrefs

Cf. A253283, A303699.

Sequence in context: A161960 A287428 A232933 * A196837 A275212 A370554

Adjacent sequences: A303697 A303698 A303699 * A303701 A303702 A303703

Keyword

sign,tabl

Author

Seiichi Manyama, Apr 28 2018

Status

approved