A290696 - OEIS

%I #4 Aug 26 2017 08:21:27

%S 1,0,0,1,0,0,1,-4,4,0,0,1,-12,48,-72,36,0,0,1,-28,268,-1056,1968,

%T -1728,576,0,0,1,-60,1200,-9480,37140,-79200,93600,-57600,14400,0,0,1,

%U -124,4924,-70080,488640,-1909440,4466880,-6393600,5486400,-2592000,518400

%N Triangle read by rows, T(n, k) = [x^k](Sum_{k=0..n}(-1)^(n-k)*Stirling2(n, k)*k!* x^k)^2, for 0 <= k <= 2n.

%C Without squaring the sum in the definition one gets for the polynomials:

%C Integral_{x=0..1} P(n, x) = Bernoulli(n, 1) = A164555(n)/A027642(n).

%F Integral_{x=0..1} P(n, x) = BernoulliMedian(n) = A212196(n)/A181131(n).

%e Triangle starts:

%e [1]

%e [0, 0, 1]

%e [0, 0, 1,  -4,    4]

%e [0, 0, 1, -12,   48,   -72,    36]

%e [0, 0, 1, -28,  268, -1056,  1968,  -1728,   576]

%e [0, 0, 1, -60, 1200, -9480, 37140, -79200, 93600, -57600, 14400]

%e The first few polynomials:

%e P_0(x) = 1

%e P_1(x) = x^2

%e P_2(x) = x^2 -  4*x^3 +   4*x^4

%e P_3(x) = x^2 - 12*x^3 +  48*x^4 -   72*x^5 +   36*x^6

%e P_4(x) = x^2 - 28*x^3 + 268*x^4 - 1056*x^5 + 1968*x^6 - 1728*x^7 + 576*x^8

%p P := (n, x) -> add((-1)^(n-k)*Stirling2(n,k)*k!*x^k, k=0..n)^2;

%p for n from 0 to 6 do seq(coeff(P(n, x), x, k), k=0..2*n) od;

%Y Cf. A278075, A291447/A291448, A212196/A181131.

%K sign,tabf

%O 0,8

%A _Peter Luschny_, Aug 25 2017