A260613 Triangle read by rows: T(n, k) = coefficient of x^(n-k) in Product_{m=1..n} (x+prime(m)); 0 <= k <= n, n >= 0.

1, 1, 2, 1, 5, 6, 1, 10, 31, 30, 1, 17, 101, 247, 210, 1, 28, 288, 1358, 2927, 2310, 1, 41, 652, 5102, 20581, 40361, 30030, 1, 58, 1349, 16186, 107315, 390238, 716167, 510510, 1, 77, 2451, 41817, 414849, 2429223, 8130689, 14117683, 9699690

Offset: 0

Author

Matthew Campbell, Aug 10 2015

Keywords

• nonn
• tabl

Comments

Up to signs and order of coefficients the same as A070918. Except for signs and the first column the same as A238146. - M. F. Hasler, Aug 13 2015

Examples

			The triangle starts:
Row 0: 1;
Row 1: 1, 2;  Coefficients of x + 2.
Row 2: 1, 5, 6;  Coefficients of (x+2)(x+3) = x^2 + 5x + 6.
Row 3: 1, 10, 31, 30; Coeff's of (x+2)(x+3)(x+5) = x^3 + 10x^2 + 31x + 30.
Row 5: 1, 17, 101, 247, 210;
Row 6: 1, 28, 288, 1358, 2927, 2310;
...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened (first 20 rows from Matthew Campbell)

Crossrefs

Main diagonal gives A002110.

Row sums give A054640.

Cf. A000040.

Programs

Maple

T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n))(mul(x+ithprime(i), i=1..n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Aug 18 2019

Mathematica

row[n_] := CoefficientList[Product[x + Prime[m], {m, 1, n}] + O[x]^(n+1), x] // Reverse;
row /@ Range[0, 8] // Flatten (* Jean-François Alcover, Sep 16 2019 *)

PARI

tabl(nn) = {for (n=0, nn, polp = prod(k=1, n, x+prime(k)); forstep (k= n, 0, -1, print1(polcoeff(polp, k), ", ");); print(););} \\ Michel Marcus, Aug 10 2015

Formula

T(n, 1) = A007504(n) for n >= 1.

T(n, 2) = A024447(n) for n >= 2.

Extensions

Corrected and edited by M. F. Hasler, Aug 13 2015

a(20) in b-file corrected by Andrew Howroyd, Dec 31 2017