A118708 Triangle, T(n, k) is the coefficient of x^k in ( Product_{j=1..n} (1 - A002110(j)*x) ), read by rows.

1, 1, -2, 1, -8, 12, 1, -38, 252, -360, 1, -248, 8232, -53280, 75600, 1, -2558, 581112, -19069200, 123152400, -174636000, 1, -32588, 77397852, -17469862560, 572771228400, -3698441208000, 5244319080000, 1, -543098, 16713897732, -39529847287080, 8919112306734000, -292409138251692000, 1888096465415160000, -2677277333530800000

Offset

0,3

Links

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Formula

T(n, k) = [x^k]( Product_{j=1..n} (1 - p(j)*x) ), where p(n) = Prime(n)*p(n-1) and p(1) = 2.

T(n, n) = A006939(n).

Example

Triangle begins as:
  1;
  1,    -2;
  1,    -8,     12;
  1,   -38,    252,      -360;
  1,  -248,   8232,    -53280,     75600;
  1, -2558, 581112, -19069200, 123152400, -174636000;

Mathematica

p[n_]:= p[n]= If[n==1, 2, Prime[n]*p[n-1]]; (* p = A002110 *)
Table[CoefficientList[Product[1 - p[j]*x, {j, n}], x], {n, 0, 12}]

Prog

(Magma)
m:=15;
function A002110(n)
  if n eq 1 then return 2;
  else return NthPrime(n)*A002110(n-1);
  end if; return A002110;
end function;
f:= func< n, x | n eq 0 select 1 else (&*[(1 - A002110(j)*x): j in [1..n]]) >;
R<x>:=PowerSeriesRing(Integers(), m+2);
T:= func< n | Coefficients(R!( f(n, x) )) >;
[T(n): n in [0..m]]; // G. C. Greubel, Dec 09 2022
(SageMath)
def p(n): return 2 if (n==1) else nth_prime(n)*p(n-1) # p = A002110
def f(n, x): return product((1 - p(j)*x) for j in range(1, n+1))
def A118708(n, k): return 1 if (n==0) else ( f(n, x) ).series(x, n+1).list()[k]
flatten([[A118708(n, k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Dec 09 2022

Crossrefs

Cf. A002110, A006939, A034386.

Sequence in context: A160485 A328821 A367382 * A055134 A137370 A214272

Adjacent sequences: A118705 A118706 A118707 * A118709 A118710 A118711

Keyword

sign,tabl

Author

Roger L. Bagula, May 20 2006

Extensions

Edited by G. C. Greubel, Dec 09 2022

Status

approved