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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":"dc1ff81e7b4d92e75fc927d6e18a96d250998c262ed783d7112575d971040760","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-69a7d6f31c3b5900","role":"sequence","name":"derivative of Even row Pascal-square read by antidiagonals.","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:   6;\n  3:  20,   24;\n  4:  42,  120,    60;\n  5:  72,  336,   420,   120;\n  6: 110,  720,  1512,  1120,   210;\n  7: 156, 1320,  3960,  5040,  2520,   336;\n  8: 210, 2184,  8580, 15840, 13860,  5040,  504;\n  9: 272, 3360, 16380, 40040, 51480, 33264, 9240, 720;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-14*n+12)*P(n,x) = (4*n^2-6*n+2+(12*n^2-30*n+6)*x)*P(n-1,x) - ((2*n-1)*x+(8*n-4)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 6*n + 2 + (12*n^2 - 30*n + 6)*x)*P[n - 1, x] - ((2*n - 1)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12)]];\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] = 6; for(n = 3, N, my(i = n + 1); v[i] = ((4*n^2 - 6*n + 2 + (12*n^2 - 30*n + 6)*x)*v[i - 1] - ((2*n - 1)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12)); 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-79ebafb266da3b41","steps":["D","D"],"edges":["e-032b739c94c6fe41","e-fb3c8d74b98903ae"],"text":"A172431 \u2192 D \u2192 D"},"headline":"A172431 \u2192 D \u2192 D","chain":null}