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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":"41f20f48e619013cacd683d6d6ceff324ac8f4b27b3b411e619519c555d01a92","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0e9006d690eb01e9","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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:      4;\n   3:     28,      32;\n   4:    124,     320,     192;\n   5:    460,    1920,    2496,     1024;\n   6:   1560,    9120,   18816,    16384,     5120;\n   7:   5016,   37920,  108672,   148480,    97280,    24576;\n   8:  15572,  144768,  534720,  1008640,  1029120,   540672,   114688;\n   9:  47156,  520960, 2364096,  5729280,  8033280,  6537216,  2867200,   524288;\n  10: 140180, 1795360, 9683328, 28787712, 51717120, 57630720, 38993920, 14680064, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-28*n+24)*P(n,x) = (12*n^2-8*n+(16*n^2-16*n)*x)*P(n-1,x) + (5*n-5+(24*n-24)*x+(16*n-16)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 8*n + (16*n^2 - 16*n)*x)*P[n - 1, x] + (5*n - 5 + (24*n - 24)*x + (16*n - 16)*x^2)*D[P[n - 1, x], x])/(8*n^2 - 28*n + 24)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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