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).

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.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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

Crossrefs

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

Sequence in context: A326952 A109909 A254573 * A030768 A051480 A071572

Adjacent sequences: A144384 A144385 A144386 * A144388 A144389 A144390

Keyword

sign,tabl

Author

Roger L. Bagula and Gary W. Adamson, Oct 01 2008

Extensions

Edited, new name, and offset corrected by Franck Maminirina Ramaharo, Oct 19 2018

Status

approved