A295516 - OEIS

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

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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:

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