# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a295516 Showing 1-1 of 1 %I A295516 #9 Dec 18 2017 04:15:41 %S A295516 1,2,-1,-2,5,-1,9,7,-11,1,-69,100,20,-28,1,-170,1049,-776,-76,94,-1, %T A295516 -1158,-4041,11573,-5612,-462,421,-1,-15533,26993,70183,-119881,41889, %U A295516 3767,-2379,1,51733,-560160,350296,1110160,-1265934,335008,35296,-16080,1 %N A295516 Triangle read by rows, T(n, k) the coefficients of some polynomials in Pi, for n >= 0 and 0 <= k <= n. %F A295516 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!. %e A295516 The polynomials in Pi start: %e A295516 1 %e A295516 2 - Pi %e A295516 -2 + 5*Pi - Pi^2 %e A295516 9 + 7*Pi - 11*Pi^2 + Pi^3 %e A295516 -69 + 100*Pi + 20*Pi^2 - 28*Pi^3 + Pi^4 %e A295516 -170 + 1049*Pi - 776*Pi^2 - 76*Pi^3 + 94*Pi^4 - Pi^5 %e A295516 The triangle starts: %e A295516 0: [ 1] %e A295516 1: [ 2, -1] %e A295516 2: [ -2, 5, -1] %e A295516 3: [ 9, 7, -11, 1] %e A295516 4: [ -69, 100, 20, -28, 1] %e A295516 5: [ -170, 1049, -776, -76, 94, -1] %e A295516 6: [-1158, -4041, 11573, -5612, -462, 421, -1] %p A295516 A295516_poly := proc(n) assume(x>-1); exp(-x)/(1 + I*log(-1-x)): series(%, x, n+1): %p A295516 simplify((I-I*Pi)^(n+1)*n!*coeff(%,x,n)); subs(I=1,%) end: %p A295516 seq(seq(coeff(A295516_poly(n), Pi, k), k=0..n), n=0..8); %K A295516 sign,tabl %O A295516 0,2 %A A295516 _Peter Luschny_, Dec 16 2017 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE