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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":"938c24a7d9a0a999651bb8b31f23db4e2649f29d6ef1ae31bb33f4f455628ecf","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-53205d2a41e99499","role":"sequence","name":"theta_plus_one of Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","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:  8;\n  2: 16,   54;\n  3: 24,  162,   192;\n  4: 32,  324,   768,   500;\n  5: 40,  540,  1920,  2500,   1080;\n  6: 48,  810,  3840,  7500,   6480,   2058;\n  7: 56, 1134,  6720, 17500,  22680,  14406,   3584;\n  8: 64, 1512, 10752, 35000,  60480,  57624,  28672,  5832;\n  9: 72, 1944, 16128, 63000, 136080, 172872, 129024, 52488, 9000;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+5*x+10*x^2+10*x^3+5*x^4+x^5)*P(n-5,x) - (5+20*x+30*x^2+20*x^3+5*x^4)*P(n-4,x) + (10+30*x+30*x^2+10*x^3)*P(n-3,x) - (10+20*x+10*x^2)*P(n-2,x) + (5+5*x)*P(n-1,x) for n >= 5, with P(0,x) = 0, P(1,x) = 8, P(2,x) = 16 + 54*x, P(3,x) = 24 + 162*x + 192*x^2, P(4,x) = 32 + 324*x + 768*x^2 + 500*x^3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 8;\nP[2, x_] := 16 + 54*x;\nP[3, x_] := 24 + 162*x + 192*x^2;\nP[4, x_] := 32 + 324*x + 768*x^2 + 500*x^3;\nP[n_, x_] := P[n, x] = Expand[(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*P[n - 5, x] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*P[n - 4, x] + (10 + 30*x + 30*x^2 + 10*x^3)*P[n - 3, x] - (10 + 20*x + 10*x^2)*P[n - 2, x] + (5 + 5*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 8; v[3] = 16 + 54*x; v[4] = 24 + 162*x + 192*x^2; v[5] = 32 + 324*x + 768*x^2 + 500*x^3; for(n = 5, N, my(i = n + 1); v[i] = (1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*v[i - 5] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*v[i - 4] + (10 + 30*x + 30*x^2 + 10*x^3)*v[i - 3] - (10 + 20*x + 10*x^2)*v[i - 2] + (5 + 5*x)*v[i - 1]); 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-e7d0ffa461d9dbc4","steps":["\u03b8+1","D"],"edges":["e-14eb908136aaeb50","e-18eda6b10e288500"],"text":"A125092 \u2192 \u03b8+1 \u2192 D"},"headline":"A125092 \u2192 \u03b8+1 \u2192 D","chain":null}