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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":"c8113326bdd294136e32947b7f4c8f07bdad17419b80f6ea2a11b36bc3df5fbf","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-a4a705640891e75c","role":"sequence","name":"veronese_section_2_0 of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd.","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:   2;\n  1:   8;\n  2:  13,    8;\n  3:  25,   56;\n  4:  33,  132,    24;\n  5:  51,  384,   240;\n  6:  62,  642,   744,    64;\n  7:  86, 1362,  2904,   832;\n  8: 100, 1970,  5910,  3200,  160;\n  9: 130, 3530, 16008, 15680, 2560;\n","recurrence_sentence":"The row polynomials satisfy 32*P(n,x) = -(706-2280*x+2208*x^2-256*x^3)*P(n-6,x) + (81-549*x+1044*x^2-576*x^3)*d/dx P(n-6,x) - (888-768*x+384*x^2)*P(n-4,x) - (162+36*x-576*x^2)*d/dx P(n-4,x) + (168+192*x)*P(n-2,x) + (81-144*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 2, P(1,x) = 8, P(2,x) = 13 + 8*x, P(3,x) = 25 + 56*x, P(4,x) = 33 + 132*x + 24*x^2, P(5,x) = 51 + 384*x + 240*x^2.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 8;\nP[2, x_] := 13 + 8*x;\nP[3, x_] := 25 + 56*x;\nP[4, x_] := 33 + 132*x + 24*x^2;\nP[5, x_] := 51 + 384*x + 240*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(706 - 2280*x + 2208*x^2 - 256*x^3)*P[n - 6, x] + (81 - 549*x + 1044*x^2 - 576*x^3)*D[P[n - 6, x], x] - (888 - 768*x + 384*x^2)*P[n - 4, x] - (162 + 36*x - 576*x^2)*D[P[n - 4, x], x] + (168 + 192*x)*P[n - 2, x] + (81 - 144*x)*D[P[n - 2, x], x])/(32)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 8; v[3] = 13 + 8*x; v[4] = 25 + 56*x; v[5] = 33 + 132*x + 24*x^2; v[6] = 51 + 384*x + 240*x^2; for(n = 6, N, my(i = n + 1); v[i] = (-(706 - 2280*x + 2208*x^2 - 256*x^3)*v[i - 6] + (81 - 549*x + 1044*x^2 - 576*x^3)*deriv(v[i - 6], x) - (888 - 768*x + 384*x^2)*v[i - 4] - (162 + 36*x - 576*x^2)*deriv(v[i - 4], x) + (168 + 192*x)*v[i - 2] + (81 - 144*x)*deriv(v[i - 2], x))/(32)); 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-5a70a6e97131d437","steps":["Veronese 2","D"],"edges":["e-4995ac39f8921064","e-f38c3bb9b4fbb42e"],"text":"A064861 \u2192 Veronese 2 \u2192 D"},"headline":"A064861 \u2192 Veronese 2 \u2192 D","chain":null}