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 A287314 Triangle read by rows, the coefficients of the polynomials generating the columns of A287316. 6
 1, 0, 1, 0, -1, 2, 0, 4, -9, 6, 0, -33, 82, -72, 24, 0, 456, -1225, 1250, -600, 120, 0, -9460, 27041, -30600, 17700, -5400, 720, 0, 274800, -826336, 1011017, -661500, 249900, -52920, 5040, 0, -10643745, 33391954, -43471624, 31149496, -13524000, 3622080, -564480, 40320 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS The zeta polynomials for the poset P_n of ordered pairs (S,T) where S,T are subsets of [n] with |S| = |T| ordered component-wise by inclusion. - Geoffrey Critzer, Jan 22 2021 LINKS FORMULA Sum_{k=0..n} abs(T(n,k)) = A000275(n) = A212855_row(2). EXAMPLE Triangle starts: [0] 1 [1] 0,      1 [2] 0,     -1,       2 [3] 0,      4,      -9,       6 [4] 0,    -33,      82,     -72,      24 [5] 0,    456,   -1225,    1250,    -600,    120 [6] 0,  -9460,   27041,  -30600,   17700,  -5400,    720 [7] 0, 274800, -826336, 1011017, -661500, 249900, -52920, 5040 ... For example let p4(x) = -33*x + 82*x^2 - 72*x^3 + 24*x^4 then p4(n) = A169712(n). MAPLE A287314_row := proc(n) local k; sum(z^k/k!^2, k = 0..infinity); series(%^x, z=0, n+1): n!^2*coeff(%, z, n); seq(coeff(%, x, k), k=0..n) end: for n from 0 to 8 do print(A287314_row(n)) od; A287314_poly := proc(n) local k, x; sum(z^k/k!^2, k = 0..infinity); series(%^x, z=0, n+1): unapply(n!^2*coeff(%, z, n), x) end: for n from 0 to 7 do A287314_poly(n) od; MATHEMATICA nn = 10; e[x_] := Sum[x^n/n!^2, {n, 0, nn}]; f[list_] := CoefficientList[InterpolatingPolynomial[Table[{i, list[[i]]}, {i, 1, nn}], m], m]; Drop[Map[f, Transpose[Table[Table[n!^2, {n, 0, nn}] CoefficientList[ Series[e[x]^k, {x, 0, nn}], x], {k, 1, nn}]]], -1] // Grid (* Geoffrey Critzer, Jan 22 2021 *) CROSSREFS Cf. A287316, A000384 (p2), A169711 (p3), A169712 (p4), A169713 (p5). Cf. A000275(n), A212855. Sequence in context: A195287 A070015 A021492 * A077119 A002938 A111938 Adjacent sequences:  A287311 A287312 A287313 * A287315 A287316 A287317 KEYWORD sign,tabl,changed AUTHOR Peter Luschny, May 27 2017 STATUS approved

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Last modified January 28 03:36 EST 2021. Contains 340490 sequences. (Running on oeis4.)