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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":"4410902ca8ad6621b33c43b7065a8468a567138053e318b1013b81e95872dcf7","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-940e4d08e5526f20","role":"sequence","name":"reversal of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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:  4;\n  2:  8,   26;\n  3: 12,   84,   120;\n  4: 16,  174,   552,   484;\n  5: 20,  296,  1488,  2924,   1820;\n  6: 24,  450,  3120,  9816,  13680,   6558;\n  7: 28,  636,  5640, 24680,  55320,  59064,   22960;\n  8: 32,  854,  9240, 52060, 163120, 280428,  241136,  78728;\n  9: 36, 1104, 14112, 97524, 395220, 952344, 1317456, 944808, 265716;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-6*n+2+(2*n^2-6*n+4)*x)*P(n,x) = (4*n^2-2*n+(26*n^2-31*n-3)*x+(20*n^2-44*n+8)*x^2+(4*n^2-12*n+8)*x^3)*P(n-1,x) - (2*n*x+(17*n-1)*x^2+(16*n-8)*x^3+(4*n-4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n + (26*n^2 - 31*n - 3)*x + (20*n^2 - 44*n + 8)*x^2 + (4*n^2 - 12*n + 8)*x^3)*P[n - 1, x] - (2*n*x + (17*n - 1)*x^2 + (16*n - 8)*x^3 + (4*n - 4)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 6*n + 2 + (2*n^2 - 6*n + 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 2*n + (26*n^2 - 31*n - 3)*x + (20*n^2 - 44*n + 8)*x^2 + (4*n^2 - 12*n + 8)*x^3)*v[i - 1] - (2*n*x + (17*n - 1)*x^2 + (16*n - 8)*x^3 + (4*n - 4)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 6*n + 2 + (2*n^2 - 6*n + 4)*x)); 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-6466a8550c08a52f","steps":["reverse","D"],"edges":["e-18c37bd14c5a739f","e-8c221de13a6ff1be"],"text":"A116414 \u2192 reverse \u2192 D"},"headline":"A116414 \u2192 reverse \u2192 D","chain":null}