%I #12 Jul 15 2021 04:37:11

%S 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,

%T 12,2,-11,29,-43,43,-25,17,7,2,-13,40,-72,86,-68,42,-10,16,2,-15,53,

%U -112,158,-154,110,-52,26,13,2,-17,68,-165,270,-312,264,-162,78,-13,18

%N 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).

%C Row sums yield the primes A000040.

%H G. C. Greubel, <a href="/A144387/b144387.txt">Rows n = 0..50 of the triangle, flattened</a>

%e Triangle begins

%e 2;

%e 2, 1;

%e 2, -1, 4;

%e 2, -3, 5, 3;

%e 2, -5, 8, -2, 8;

%e 2, -7, 13, -10, 10, 5;

%e 2, -9, 20, -23, 20, -5, 12;

%e 2, -11, 29, -43, 43, -25, 17, 7;

%e 2, -13, 40, -72, 86, -68, 42, -10, 16;

%e 2, -15, 53, -112, 158, -154, 110, -52, 26, 13;

%e 2, -17, 68, -165, 270, -312, 264, -162, 78, -13, 18;

%e ...

%t p[x_, n_] = Sum[Prime[k + 1]*x^k*(1 - x)^(n - k), {k, 0, n}];

%t Table[CoefficientList[p[x, n], x], {n, 0, 10}]//Flatten

%o (Sage)

%o def p(n,x): return sum( nth_prime(j+1)*x^j*(1-x)^(n-j) for j in (0..n) )

%o def T(n): return ( p(n,x) ).full_simplify().coefficients(sparse=False)

%o [T(n) for n in (0..12)] # _G. C. Greubel_, Jul 15 2021

%Y Cf. A122753, A123018, A123019, A123021, A123027, A123199, A123202, A123217, A123221, A141720, A144400, A174128.

%K sign,tabl

%O 0,1

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 01 2008

%E Edited, new name, and offset corrected by _Franck Maminirina Ramaharo_, Oct 19 2018