A295516 Triangle read by rows, T(n, k) the coefficients of some polynomials in Pi, for n >= 0 and 0 <= k <= n.

1, 2, -1, -2, 5, -1, 9, 7, -11, 1, -69, 100, 20, -28, 1, -170, 1049, -776, -76, 94, -1, -1158, -4041, 11573, -5612, -462, 421, -1, -15533, 26993, 70183, -119881, 41889, 3767, -2379, 1, 51733, -560160, 350296, 1110160, -1265934, 335008, 35296, -16080, 1

Offset

0,2

Links

Table of n, a(n) for n=0..44.

Formula

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!.

Example

The polynomials in Pi start:
                               1
                             2 - Pi
                        -2 + 5*Pi - Pi^2
                    9 + 7*Pi - 11*Pi^2  + Pi^3
              -69 + 100*Pi + 20*Pi^2  - 28*Pi^3  + Pi^4
        -170 + 1049*Pi - 776*Pi^2  - 76*Pi^3  + 94*Pi^4  - Pi^5
The triangle starts:
0: [    1]
1: [    2,    -1]
2: [   -2,     5,    -1]
3: [    9,     7,   -11,     1]
4: [  -69,   100,    20,   -28,    1]
5: [ -170,  1049,  -776,   -76,   94,  -1]
6: [-1158, -4041, 11573, -5612, -462, 421, -1]

Maple

A295516_poly := proc(n) assume(x>-1); exp(-x)/(1 + I*log(-1-x)): series(%, x, n+1):
simplify((I-I*Pi)^(n+1)*n!*coeff(%, x, n)); subs(I=1, %) end:
seq(seq(coeff(A295516_poly(n), Pi, k), k=0..n), n=0..8);

Crossrefs

Sequence in context: A225568 A291694 A135506 * A330209 A068822 A351517

Adjacent sequences: A295513 A295514 A295515 * A295517 A295518 A295519

Keyword

sign,tabl

Author

Peter Luschny, Dec 16 2017

Status

approved