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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":"f69f14babe9c2974f48cd9e384e81f20104387b66062744129b40773a58afb9a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e5bdf85b655e989a","role":"sequence","name":"shift_x_plus_one of Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).","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:     0;\n  2:     4;\n  3:    16;\n  4:    72,    32;\n  5:   272,   192;\n  6:  1004,  1056,   192;\n  7:  3552,  4864,  1536;\n  8: 12304, 20928,  9984,  1024;\n  9: 41824, 84864, 53760, 10240;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = (5*n-11+(4*n-12)*x)*P(n-1,x) - (6+14*x+8*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n - 11 + (4*n - 12)*x)*P[n - 1, x] - (6 + 14*x + 8*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 4; for(n = 3, N, my(i = n + 1); v[i] = ((5*n - 11 + (4*n - 12)*x)*v[i - 1] - (6 + 14*x + 8*x^2)*deriv(v[i - 1], x))/(n - 2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cf38558d99f4fe37","steps":["x \u21a6 x+1","D"],"edges":["e-7b35457a4f9e8cf8","e-375b869984275436"],"text":"A229032 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A229032 \u2192 x \u21a6 x+1 \u2192 D","chain":null}