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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":"7a190555a0fc42c4cf0cb79e54261ce513eee046463ca28af3c1e2e2c45f2a16","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6321b10e19f62245","role":"sequence","name":"basis_map_A013609 of Riordan array (1, 3+x).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      0;\n   2:      6;\n   3:     38,       72;\n   4:    186,      696,      648;\n   5:    814,     4544,     8424,      5184;\n   6:   3348,    24840,    68904,     84672,     38880;\n   7:  13236,   122496,   452328,    832896,    764640,    279936;\n   8:  50910,   564552,  2602872,   6385728,   8791200,   6438528,   1959552;\n   9: 191906,  2480016, 13709400,  42019264,  77113440,  84727296,  51601536,  13436928;\n  10: 712302, 10511112, 67744872, 249059712, 571240320, 836936064, 764878464, 398628864, 90699264;\n","recurrence_sentence":"The row polynomials satisfy (6*n^2-72*n+120-(108*n-216)*x)*P(n,x) = -(4*n^2-114*n+572-(180*n-1728)*x+1296*x^2)*P(n-1,x) + (13*n-143+(44*n-718)*x+(36*n-1188)*x^2-648*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 6.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 114*n + 572 - (180*n - 1728)*x + 1296*x^2)*P[n - 1, x] + (13*n - 143 + (44*n - 718)*x + (36*n - 1188)*x^2 - 648*x^3)*D[P[n - 1, x], x])/(6*n^2 - 72*n + 120 - (108*n - 216)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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