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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":"a1a48a528b1c4dc6f55ecb124242a2944606f39ebc3be2074c7555c164efc7ba","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-dc3e6727bddaa3d6","role":"sequence","name":"derivative of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:   1;\n  3:   5,    2;\n  4:  16,   16,    3;\n  5:  40,   72,   33,    4;\n  6:  85,  238,  192,   56,    5;\n  7: 161,  644,  792,  400,   85,    6;\n  8: 280, 1512, 2610, 1980,  720,  120,   7;\n  9: 456, 3192, 7326, 7700, 4160, 1176, 161, 8;\n","recurrence_sentence":"The row polynomials satisfy (2*n^4-13*n^3+39*n^2-60*n+36-(2*n^2-7*n+6)*x^2)*P(n,x) = (2*n^4-5*n^3+11*n^2-12*n+(n^4-2*n^3+n^2)*x-(3*n^2-4*n)*x^2-(n^2-n)*x^3)*P(n-1,x) + ((4*n^3-12*n^2+16*n-8)*x+(n^3-3*n^2-4*n+6)*x^2-(6*n-6)*x^3-(n-1)*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^4 - 5*n^3 + 11*n^2 - 12*n + (n^4 - 2*n^3 + n^2)*x - (3*n^2 - 4*n)*x^2 - (n^2 - n)*x^3)*P[n - 1, x] + ((4*n^3 - 12*n^2 + 16*n - 8)*x + (n^3 - 3*n^2 - 4*n + 6)*x^2 - (6*n - 6)*x^3 - (n - 1)*x^4)*D[P[n - 1, x], x])/(2*n^4 - 13*n^3 + 39*n^2 - 60*n + 36 - (2*n^2 - 7*n + 6)*x^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] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^4 - 5*n^3 + 11*n^2 - 12*n + (n^4 - 2*n^3 + n^2)*x - (3*n^2 - 4*n)*x^2 - (n^2 - n)*x^3)*v[i - 1] + ((4*n^3 - 12*n^2 + 16*n - 8)*x + (n^3 - 3*n^2 - 4*n + 6)*x^2 - (6*n - 6)*x^3 - (n - 1)*x^4)*deriv(v[i - 1], x))/(2*n^4 - 13*n^3 + 39*n^2 - 60*n + 36 - (2*n^2 - 7*n + 6)*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-0eba401b117f16a9","steps":["D","D+1"],"edges":["e-04abbda3a865d9ca","e-a7acbdd4acfba291"],"text":"A085478 \u2192 D \u2192 D+1"},"headline":"A085478 \u2192 D \u2192 D+1","chain":null}