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

%I #12 Dec 09 2022 07:10:39

%S 1,1,-2,1,-8,12,1,-38,252,-360,1,-248,8232,-53280,75600,1,-2558,

%T 581112,-19069200,123152400,-174636000,1,-32588,77397852,-17469862560,

%U 572771228400,-3698441208000,5244319080000,1,-543098,16713897732,-39529847287080,8919112306734000,-292409138251692000,1888096465415160000,-2677277333530800000

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

%H G. C. Greubel, <a href="/A118708/b118708.txt">Rows n = 0..30 of the triangle, flattened</a>

%F 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.

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

%e Triangle begins as:

%e 1;

%e 1, -2;

%e 1, -8, 12;

%e 1, -38, 252, -360;

%e 1, -248, 8232, -53280, 75600;

%e 1, -2558, 581112, -19069200, 123152400, -174636000;

%t p[n_]:= p[n]= If[n==1, 2, Prime[n]*p[n-1]]; (* p = A002110 *)

%t Table[CoefficientList[Product[1 - p[j]*x, {j, n}], x], {n, 0, 12}]

%o (Magma)

%o m:=15;

%o function A002110(n)

%o if n eq 1 then return 2;

%o else return NthPrime(n)*A002110(n-1);

%o end if; return A002110;

%o end function;

%o f:= func< n, x | n eq 0 select 1 else (&*[(1 - A002110(j)*x): j in [1..n]]) >;

%o R<x>:=PowerSeriesRing(Integers(), m+2);

%o T:= func< n | Coefficients(R!( f(n,x) )) >;

%o [T(n): n in [0..m]]; // _G. C. Greubel_, Dec 09 2022

%o (SageMath)

%o def p(n): return 2 if (n==1) else nth_prime(n)*p(n-1) # p = A002110

%o def f(n, x): return product((1 - p(j)*x) for j in range(1,n+1))

%o def A118708(n,k): return 1 if (n==0) else ( f(n,x) ).series(x,n+1).list()[k]

%o flatten([[A118708(n,k) for k in range(n+1)] for n in range(16)]) # _G. C. Greubel_, Dec 09 2022

%Y Cf. A002110, A006939, A034386.

%K sign,tabl

%O 0,3

%A _Roger L. Bagula_, May 20 2006

%E Edited by _G. C. Greubel_, Dec 09 2022