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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"cbd41881e117ab61affe7ca705164a24f40d46f104050276da134c7c7d2fefb3","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-9189beb5ff99695e","role":"sequence","name":"Triangle T(n, k) = (m*(n-k) + 1)*T(n-1, k-1) + (m*(k-1) + 1)*T(n-1, k) + j*T(n-2, k-1), where T(n, 1) = T(n, n) = 1, m = 1, and j = 4, read by rows.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    0;\n  1:    1;\n  2:    8,      2;\n  3:   23,     46,       3;\n  4:   54,    340,     162,        4;\n  5:  117,   1636,    2454,      468,        5;\n  6:  244,   6510,   21672,    13020,     1220,       6;\n  7:  499,  23394,  145329,   193772,    58485,    2994,      7;\n  8: 1010,  79120,  830922,  2053304,  1384870,  237360,   7070,     8;\n  9: 2033, 257512, 4294812, 17688984, 22111230, 8589624, 901292, 16264, 9;\n","recurrence_sentence":"The row polynomials satisfy (n+3)*P(n,x) = -(16*n+64)*x^2*P(n-4,x) - ((8*n+32)*x+(8*n^2+16*n-48)*x^2)*P(n-3,x) - ((4*n+16)*x^2-(4*n+16)*x^3)*d/dx P(n-3,x) - (2*n+8+(2*n^2-4*n-40)*x+(n^3+n^2-6*n)*x^2)*P(n-2,x) - ((n+4)*x+(n^2+2*n-4)*x^2-(n^2+3*n)*x^3)*d/dx P(n-2,x) + (3*n+10+(2*n^2+4*n-6)*x)*P(n-1,x) + ((n+3)*x-(n+3)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 2*x, P(3,x) = 23 + 46*x + 3*x^2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 8 + 2*x;\nP[3, x_] := 23 + 46*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(16*n + 64)*x^2*P[n - 4, x] - ((8*n + 32)*x + (8*n^2 + 16*n - 48)*x^2)*P[n - 3, x] - ((4*n + 16)*x^2 - (4*n + 16)*x^3)*D[P[n - 3, x], x] - (2*n + 8 + (2*n^2 - 4*n - 40)*x + (n^3 + n^2 - 6*n)*x^2)*P[n - 2, x] - ((n + 4)*x + (n^2 + 2*n - 4)*x^2 - (n^2 + 3*n)*x^3)*D[P[n - 2, x], x] + (3*n + 10 + (2*n^2 + 4*n - 6)*x)*P[n - 1, x] + ((n + 3)*x - (n + 3)*x^2)*D[P[n - 1, x], x])/(n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); 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