A295517 - OEIS

%I #5 Dec 18 2017 04:15:49

%S 1,2,-1,6,-5,-1,27,-31,-11,1,167,-252,-136,28,1,1310,-2491,-1864,656,

%T 94,-1,12394,-28603,-27583,13952,3718,-421,-1,137053,-372765,-440425,

%U 290431,113119,-24739,-2379,1,1733325,-5433312,-7596496,6162480,3142746,-1010144,-189768,16080,1

%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 + 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 The first few polynomials are:

%e                                1

%e                              2 - Pi

%e                          6 - 5 Pi - Pi^2

%e                    27 - 31 Pi - 11 Pi^2  + Pi^3

%e              167 - 252 Pi - 136 Pi^2  + 28 Pi^3  + Pi^4

%e        1310 - 2491 Pi - 1864 Pi^2  + 656 Pi^3  + 94 Pi^4  - Pi^5

%e 12394 - 28603 Pi - 27583 Pi^2 + 13952 Pi^3 + 3718 Pi^4 - 421 Pi^5 - Pi^6

%e The triangle starts:

%e 0:      1

%e 1:      2,      -1

%e 2:      6,      -5,      -1

%e 3:     27,     -31,     -11,      1

%e 4:    167,    -252,    -136,     28,      1

%e 5:   1310,   -2491,   -1864,    656,     94,     -1

%e 6:  12394,  -28603,  -27583,  13952,   3718,   -421,    -1

%e 7: 137053, -372765, -440425, 290431, 113119, -24739, -2379, 1

%p A295517_poly := proc(n) assume(x<-1); exp(-x)/(1 + log(-1-x)): series(%, x, n+1):

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

%p seq(seq(coeff(A295517_poly(n), Pi, k), k=0..n), n=0..8);

%Y Cf. A291979.

%K sign,tabl

%O 0,2

%A _Peter Luschny_, Dec 17 2017