A327809 Regular triangle, coefficients of the polynomial P(n)(x) = (-1)^(n+1)*(2*n+1)*binomial(2*n, n)*Sum_{i=0..n} x^i*binomial(n, i)/(n+i+1).

-1, 3, 2, -10, -15, -6, 35, 84, 70, 20, -126, -420, -540, -315, -70, 462, 1980, 3465, 3080, 1386, 252, -1716, -9009, -20020, -24024, -16380, -6006, -924, 6435, 40040, 108108, 163800, 150150, 83160, 25740, 3432, -24310, -175032, -556920, -1021020, -1178100, -875160, -408408, -109395, -12870

Offset

0,2

Links

Table of n, a(n) for n=0..44.

Karl Dilcher, Maciej Ulas, Arithmetic properties of polynomial solutions of the Diophantine equation P(x)x^(n+1)+Q(x)(x+1)^(n+1)=1, arXiv:1909.11222 [math.NT], 2019. See Pn(x) Table 1 p. 2.

Example

Triangle begins:
     -1;
      3,     2;
    -10,   -15,     -6;
     35,    84,     70,     20;
   -126,  -420,   -540,   -315,   -70;
    462,  1980,   3465,   3080,   1386,   252;
  -1716, -9009, -20020, -24024, -16380, -6006, -924;
  ...

Prog

(PARI) pol(n) = (-1)^(n+1)*(2*n+1)*binomial(2*n, n)*sum(i=0, n, x^i*binomial(n, i)/(n+i+1));
row(n) = Vecrev(pol(n));
tabl(nn) = for (n=0, nn, print(row(n)));

Crossrefs

Cf. A046899 (Q(x) polynomials, up to sign).

Cf. A001700 (1st column, up to sign), A033876 (right diagonal, up to sign).

Sequence in context: A256063 A006743 A091811 * A075856 A025520 A099946

Adjacent sequences: A327806 A327807 A327808 * A327810 A327811 A327812

Keyword

sign,tabl

Author

Michel Marcus, Sep 26 2019

Status

approved