A290696 - OEIS

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

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OFFSET

0,8

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).

LINKS

Table of n, a(n) for n=0..48.

FORMULA

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

EXAMPLE

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

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;

CROSSREFS

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

Sequence in context: A189973 A282866 A098445 * A200515 A200505 A285242

Adjacent sequences: A290693 A290694 A290695 * A290697 A290698 A290699

KEYWORD

sign,tabf

AUTHOR

Peter Luschny, Aug 25 2017

STATUS

approved