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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":"a21a6c78c55c845454fe2286e054c4aa137443890952bd6d22a10b010f59ca66","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-a2304b34e998691c","role":"sequence","name":"Triangle read by rows: T(n, k) = [z^k] P(n, z) where P(n, z) = Sum_{k=0..n} binomial(n, k) * Pochhammer(n - k + c, k) * z^k / k! and c = 4.","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:   1;\n   2:   5,    4;\n   3:  11,   30,    10;\n   4:  19,  108,   105,     20;\n   5:  29,  280,   546,    280,     35;\n   6:  41,  600,  1960,   1960,    630,     56;\n   7:  55, 1134,  5580,   9240,   5670,   1260,     84;\n   8:  71, 1960, 13545,  33600,  33810,  14112,   2310,   120;\n   9:  89, 3168, 29260, 101640, 152460, 103488,  31416,  3960,  165;\n  10: 109, 4860, 57816, 267960, 561330, 565488, 277200, 64152, 6435, 220;\n","recurrence_sentence":"The row polynomials satisfy (n^4-4*n^2+n+2+(2*n^3-n^2-7*n+6)*x+(n^2+n-2)*x^2)*P(n,x) = (n^4+2*n^3-2*n^2-3*n+2+(3*n^4+2*n^3-8*n^2+n+2)*x+(6*n^3-3*n^2+3*n-6)*x^2+(3*n^2+n+2)*x^3)*P(n-1,x) + ((2*n^3+2*n^2-4*n)*x-(2*n^3-2*n^2-4*n+1)*x^2-(4*n^2-2*n-2)*x^3-(2*n+1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^4 + 2*n^3 - 2*n^2 - 3*n + 2 + (3*n^4 + 2*n^3 - 8*n^2 + n + 2)*x + (6*n^3 - 3*n^2 + 3*n - 6)*x^2 + (3*n^2 + n + 2)*x^3)*P[n - 1, x] + ((2*n^3 + 2*n^2 - 4*n)*x - (2*n^3 - 2*n^2 - 4*n + 1)*x^2 - (4*n^2 - 2*n - 2)*x^3 - (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^2 + n + 2 + (2*n^3 - n^2 - 7*n + 6)*x + (n^2 + n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); 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