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.

1, 0, 0, 1, 0, 0, 1, -4, 4, 0, 0, 1, -12, 48, -72, 36, 0, 0, 1, -28, 268, -1056, 1968, -1728, 576, 0, 0, 1, -60, 1200, -9480, 37140, -79200, 93600, -57600, 14400, 0, 0, 1, -124, 4924, -70080, 488640, -1909440, 4466880, -6393600, 5486400, -2592000, 518400

Offset: 0

Author

Peter Luschny, Aug 25 2017

Keywords

• sign
• tabf

Comments

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

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

Examples

			Triangle starts:
[1]
[0, 0, 1]
[0, 0, 1,  -4,    4]
[0, 0, 1, -12,   48,   -72,    36]
[0, 0, 1, -28,  268, -1056,  1968,  -1728,   576]
[0, 0, 1, -60, 1200, -9480, 37140, -79200, 93600, -57600, 14400]
The first few polynomials:
P_0(x) = 1
P_1(x) = x^2
P_2(x) = x^2 -  4*x^3 +   4*x^4
P_3(x) = x^2 - 12*x^3 +  48*x^4 -   72*x^5 +   36*x^6
P_4(x) = x^2 - 28*x^3 + 268*x^4 - 1056*x^5 + 1968*x^6 - 1728*x^7 + 576*x^8
		

Crossrefs

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

Programs

Maple

P := (n, x) -> add((-1)^(n-k)*Stirling2(n,k)*k!*x^k, k=0..n)^2;
for n from 0 to 6 do seq(coeff(P(n, x), x, k), k=0..2*n) od;

Formula

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