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A144387
Triangle read by rows: row n gives the coefficients in the expansion of Sum_{j=0..n} A000040(j+1)*x^j*(1 - x)^(n - j).
4
2, 2, 1, 2, -1, 4, 2, -3, 5, 3, 2, -5, 8, -2, 8, 2, -7, 13, -10, 10, 5, 2, -9, 20, -23, 20, -5, 12, 2, -11, 29, -43, 43, -25, 17, 7, 2, -13, 40, -72, 86, -68, 42, -10, 16, 2, -15, 53, -112, 158, -154, 110, -52, 26, 13, 2, -17, 68, -165, 270, -312, 264, -162, 78, -13, 18
OFFSET
0,1
COMMENTS
Row sums yield the primes A000040.
EXAMPLE
Triangle begins
2;
2, 1;
2, -1, 4;
2, -3, 5, 3;
2, -5, 8, -2, 8;
2, -7, 13, -10, 10, 5;
2, -9, 20, -23, 20, -5, 12;
2, -11, 29, -43, 43, -25, 17, 7;
2, -13, 40, -72, 86, -68, 42, -10, 16;
2, -15, 53, -112, 158, -154, 110, -52, 26, 13;
2, -17, 68, -165, 270, -312, 264, -162, 78, -13, 18;
...
MATHEMATICA
p[x_, n_] = Sum[Prime[k + 1]*x^k*(1 - x)^(n - k), {k, 0, n}];
Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten
PROG
(Sage)
def p(n, x): return sum( nth_prime(j+1)*x^j*(1-x)^(n-j) for j in (0..n) )
def T(n): return ( p(n, x) ).full_simplify().coefficients(sparse=False)
[T(n) for n in (0..12)] # G. C. Greubel, Jul 15 2021
KEYWORD
sign,tabl
AUTHOR
EXTENSIONS
Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 19 2018
STATUS
approved