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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":"9f3640a6ce61a421f3b6092ca58961a2caa3f57159475e08ce6413468d2270d7","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-ba966decf74c4020","role":"sequence","name":"gamma_polynomial of Pascal-(1,9,1) array.","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:  1;\n  1:  1;\n  2: 10,    9;\n  3: 19,   18;\n  4: 28,  189,    81;\n  5: 37,  522,   243;\n  6: 46, 1017,  2673,    729;\n  7: 55, 1674,  9558,   2916;\n  8: 64, 2493, 23085,  33534,  6561;\n  9: 73, 3474, 45441, 145800, 32805;\n","recurrence_sentence":"The row polynomials satisfy (9*n^3-26*n^2+17*n+(36*n^2-89*n)*x+36*n*x^2)*P(n,x) = (9*n^3-17*n^2+8*n+(162*n^3-594*n^2+379*n)*x+(648*n^2-1566*n+954)*x^2+(648*n-648)*x^3)*P(n-1,x) - ((9*n^2-8*n)*x+(324*n^2-252*n-17)*x^2+(1296*n-576)*x^3+1296*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 17*n^2 + 8*n + (162*n^3 - 594*n^2 + 379*n)*x + (648*n^2 - 1566*n + 954)*x^2 + (648*n - 648)*x^3)*P[n - 1, x] - ((9*n^2 - 8*n)*x + (324*n^2 - 252*n - 17)*x^2 + (1296*n - 576)*x^3 + 1296*x^4)*D[P[n - 1, x], x])/(9*n^3 - 26*n^2 + 17*n + (36*n^2 - 89*n)*x + 36*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((9*n^3 - 17*n^2 + 8*n + (162*n^3 - 594*n^2 + 379*n)*x + (648*n^2 - 1566*n + 954)*x^2 + (648*n - 648)*x^3)*v[i - 1] - ((9*n^2 - 8*n)*x + (324*n^2 - 252*n - 17)*x^2 + (1296*n - 576)*x^3 + 1296*x^4)*deriv(v[i - 1], x))/(9*n^3 - 26*n^2 + 17*n + (36*n^2 - 89*n)*x + 36*n*x^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-7ac21ea1b3b7e282","steps":["\u03b3","D+1"],"edges":["e-22858e6475490a6d","e-23cb4aba4654b3a8"],"text":"A143685 \u2192 \u03b3 \u2192 D+1"},"headline":"A143685 \u2192 \u03b3 \u2192 D+1","chain":null}