%I #9 Dec 18 2017 04:15:41

%S 1,2,-1,-2,5,-1,9,7,-11,1,-69,100,20,-28,1,-170,1049,-776,-76,94,-1,

%T -1158,-4041,11573,-5612,-462,421,-1,-15533,26993,70183,-119881,41889,

%U 3767,-2379,1,51733,-560160,350296,1110160,-1265934,335008,35296,-16080,1

%N Triangle read by rows, T(n, k) the coefficients of some polynomials in Pi, for n >= 0 and 0 <= k <= n.

%F Consider the polynomial p_n(x) with e.g.f. exp(-x)/(1 + i*log(-1-x)). After multiplying with (i-i*Pi)^(n+1) and then substituting i by 1 this becomes a polynomial in Pi, the coefficients of which in ascending order constitute row n of the triangle. The sum of the coefficients is n!.

%e The polynomials in Pi start:

%e 1

%e 2 - Pi

%e -2 + 5*Pi - Pi^2

%e 9 + 7*Pi - 11*Pi^2 + Pi^3

%e -69 + 100*Pi + 20*Pi^2 - 28*Pi^3 + Pi^4

%e -170 + 1049*Pi - 776*Pi^2 - 76*Pi^3 + 94*Pi^4 - Pi^5

%e The triangle starts:

%e 0: [ 1]

%e 1: [ 2, -1]

%e 2: [ -2, 5, -1]

%e 3: [ 9, 7, -11, 1]

%e 4: [ -69, 100, 20, -28, 1]

%e 5: [ -170, 1049, -776, -76, 94, -1]

%e 6: [-1158, -4041, 11573, -5612, -462, 421, -1]

%p A295516_poly := proc(n) assume(x>-1); exp(-x)/(1 + I*log(-1-x)): series(%, x, n+1):

%p simplify((I-I*Pi)^(n+1)*n!*coeff(%,x,n)); subs(I=1,%) end:

%p seq(seq(coeff(A295516_poly(n), Pi, k), k=0..n), n=0..8);

%K sign,tabl

%O 0,2

%A _Peter Luschny_, Dec 16 2017