cp's OEIS Frontend

A290696 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.

%I A290696 #4 Aug 26 2017 08:21:27
%S A290696 1,0,0,1,0,0,1,-4,4,0,0,1,-12,48,-72,36,0,0,1,-28,268,-1056,1968,
%T A290696 -1728,576,0,0,1,-60,1200,-9480,37140,-79200,93600,-57600,14400,0,0,1,
%U A290696 -124,4924,-70080,488640,-1909440,4466880,-6393600,5486400,-2592000,518400
%N A290696 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 A290696 Without squaring the sum in the definition one gets for the polynomials:
%C A290696 Integral_{x=0..1} P(n, x) = Bernoulli(n, 1) = A164555(n)/A027642(n).
%F A290696 Integral_{x=0..1} P(n, x) = BernoulliMedian(n) = A212196(n)/A181131(n).
%e A290696 Triangle starts:
%e A290696 [1]
%e A290696 [0, 0, 1]
%e A290696 [0, 0, 1,  -4,    4]
%e A290696 [0, 0, 1, -12,   48,   -72,    36]
%e A290696 [0, 0, 1, -28,  268, -1056,  1968,  -1728,   576]
%e A290696 [0, 0, 1, -60, 1200, -9480, 37140, -79200, 93600, -57600, 14400]
%e A290696 The first few polynomials:
%e A290696 P_0(x) = 1
%e A290696 P_1(x) = x^2
%e A290696 P_2(x) = x^2 -  4*x^3 +   4*x^4
%e A290696 P_3(x) = x^2 - 12*x^3 +  48*x^4 -   72*x^5 +   36*x^6
%e A290696 P_4(x) = x^2 - 28*x^3 + 268*x^4 - 1056*x^5 + 1968*x^6 - 1728*x^7 + 576*x^8
%p A290696 P := (n, x) -> add((-1)^(n-k)*Stirling2(n,k)*k!*x^k, k=0..n)^2;
%p A290696 for n from 0 to 6 do seq(coeff(P(n, x), x, k), k=0..2*n) od;
%Y A290696 Cf. A278075, A291447/A291448, A212196/A181131.
%K A290696 sign,tabf
%O A290696 0,8
%A A290696 _Peter Luschny_, Aug 25 2017