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

1, 2, -1, 6, -5, -1, 27, -31, -11, 1, 167, -252, -136, 28, 1, 1310, -2491, -1864, 656, 94, -1, 12394, -28603, -27583, 13952, 3718, -421, -1, 137053, -372765, -440425, 290431, 113119, -24739, -2379, 1, 1733325, -5433312, -7596496, 6162480, 3142746, -1010144, -189768, 16080, 1

Offset: 0

Author

Peter Luschny, Dec 17 2017

Keywords

• sign
• tabl

Examples

			The first few polynomials are:
                               1
                             2 - Pi
                         6 - 5 Pi - Pi^2
                   27 - 31 Pi - 11 Pi^2  + Pi^3
             167 - 252 Pi - 136 Pi^2  + 28 Pi^3  + Pi^4
       1310 - 2491 Pi - 1864 Pi^2  + 656 Pi^3  + 94 Pi^4  - Pi^5
12394 - 28603 Pi - 27583 Pi^2 + 13952 Pi^3 + 3718 Pi^4 - 421 Pi^5 - Pi^6
The triangle starts:
0:      1
1:      2,      -1
2:      6,      -5,      -1
3:     27,     -31,     -11,      1
4:    167,    -252,    -136,     28,      1
5:   1310,   -2491,   -1864,    656,     94,     -1
6:  12394,  -28603,  -27583,  13952,   3718,   -421,    -1
7: 137053, -372765, -440425, 290431, 113119, -24739, -2379, 1
		

Crossrefs

Cf. A291979.

Programs

Maple

A295517_poly := proc(n) assume(x<-1); exp(-x)/(1 + log(-1-x)): series(%, x, n+1):
simplify(-(Pi-1)^(n+1)*n!*coeff(%, x, n)); subs(I=1, %) end:
seq(seq(coeff(A295517_poly(n), Pi, k), k=0..n), n=0..8);

Formula

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.