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 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. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified April 14 10:04 EDT 2024. Contains 371657 sequences. (Running on oeis4.)