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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":"1fd99e384ef6c50cf200c9e320d07d9a1772524ec4cda6adfd9bd4c256cdc41b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1852b727a5d6b55d","role":"sequence","name":"basis_map_A013609 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:     2;\n  3:    10,     8;\n  4:    38,    64,     24;\n  5:   130,   336,    264,     64;\n  6:   420,  1464,   1776,    896,    160;\n  7:  1308,  5736,   9456,   7360,   2720,   384;\n  8:  3970, 20976,  43800,  46400,  26400,  7680,   896;\n  9: 11822, 73072, 184872, 248640, 192480, 86016, 20608, 2048;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-14*n+12)*P(n,x) = (6*n^2-8*n+(4*n^2-4*n)*x)*P(n-1,x) + (5*n-5+(12*n-12)*x+(4*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) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n^2 - 8*n + (4*n^2 - 4*n)*x)*P[n - 1, x] + (5*n - 5 + (12*n - 12)*x + (4*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] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((6*n^2 - 8*n + (4*n^2 - 4*n)*x)*v[i - 1] + (5*n - 5 + (12*n - 12)*x + (4*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-0eba401b117f16a9","steps":["basis A013609","D"],"edges":["e-354781356c20418d","e-901483510bbb101c"],"text":"A085478 \u2192 basis A013609 \u2192 D"},"headline":"A085478 \u2192 basis A013609 \u2192 D","chain":null}