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A295517 Triangle read by rows, T(n, k) the coefficients of some polynomials in Pi, for n >= 0 and 0 <= k <= n.

%I A295517 #5 Dec 18 2017 04:15:49
%S A295517 1,2,-1,6,-5,-1,27,-31,-11,1,167,-252,-136,28,1,1310,-2491,-1864,656,
%T A295517 94,-1,12394,-28603,-27583,13952,3718,-421,-1,137053,-372765,-440425,
%U A295517 290431,113119,-24739,-2379,1,1733325,-5433312,-7596496,6162480,3142746,-1010144,-189768,16080,1
%N A295517 Triangle read by rows, T(n, k) the coefficients of some polynomials in Pi, for n >= 0 and 0 <= k <= n.
%F A295517 Consider the polynomial p_n(x) with e.g.f. exp(-x)/(1 + log(-1-x)). After multiplying with -(Pi-1)^(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 constant coefficients are A291979.
%e A295517 The first few polynomials are:
%e A295517                                1
%e A295517                              2 - Pi
%e A295517                          6 - 5 Pi - Pi^2
%e A295517                    27 - 31 Pi - 11 Pi^2  + Pi^3
%e A295517              167 - 252 Pi - 136 Pi^2  + 28 Pi^3  + Pi^4
%e A295517        1310 - 2491 Pi - 1864 Pi^2  + 656 Pi^3  + 94 Pi^4  - Pi^5
%e A295517 12394 - 28603 Pi - 27583 Pi^2 + 13952 Pi^3 + 3718 Pi^4 - 421 Pi^5 - Pi^6
%e A295517 The triangle starts:
%e A295517 0:      1
%e A295517 1:      2,      -1
%e A295517 2:      6,      -5,      -1
%e A295517 3:     27,     -31,     -11,      1
%e A295517 4:    167,    -252,    -136,     28,      1
%e A295517 5:   1310,   -2491,   -1864,    656,     94,     -1
%e A295517 6:  12394,  -28603,  -27583,  13952,   3718,   -421,    -1
%e A295517 7: 137053, -372765, -440425, 290431, 113119, -24739, -2379, 1
%p A295517 A295517_poly := proc(n) assume(x<-1); exp(-x)/(1 + log(-1-x)): series(%, x, n+1):
%p A295517 simplify(-(Pi-1)^(n+1)*n!*coeff(%, x, n)); subs(I=1, %) end:
%p A295517 seq(seq(coeff(A295517_poly(n), Pi, k), k=0..n), n=0..8);
%Y A295517 Cf. A291979.
%K A295517 sign,tabl
%O A295517 0,2
%A A295517 _Peter Luschny_, Dec 17 2017