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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":"17173957e75f60edf0a7fc831650ad0665b3bbd36996c072480fb260b5309517","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-8ac1223942f7b8c5","role":"sequence","name":"reversal of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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:  1;\n  1:  5,    4;\n  2:  9,   38,    15;\n  3: 13,  104,   214,     56;\n  4: 17,  202,   789,   1068,    209;\n  5: 21,  332,  1932,   4956,   4991,     780;\n  6: 25,  494,  3835,  14664,  27926,   22378,    2911;\n  7: 29,  688,  6690,  34160,  96930,  146592,   97516,   10864;\n  8: 33,  914, 10689,  68436, 259375,  584052,  732006,  416184,   40545;\n  9: 37, 1172, 16024, 123508, 587209, 1765868, 3292668, 3521976, 1748217, 151316;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-20*n^2+6*n+(24*n^3-40*n^2+10*n)*x+(48*n^2-80*n)*x^2+24*n*x^3)*P(n,x) = (16*n^3-4*n^2-2*n+(120*n^3-92*n^2-50*n)*x+(240*n^3-264*n^2-192*n+32)*x^2+(144*n^3+48*n^2-348*n+148)*x^3+(288*n^2-336*n+144)*x^4+(144*n-144)*x^5)*P(n-1,x) - ((8*n^2+2*n)*x+(76*n^2+20*n-3)*x^2+(192*n^2+80*n-28)*x^3+(144*n^2+240*n-56)*x^4+(288*n+48)*x^5+144*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^3 - 4*n^2 - 2*n + (120*n^3 - 92*n^2 - 50*n)*x + (240*n^3 - 264*n^2 - 192*n + 32)*x^2 + (144*n^3 + 48*n^2 - 348*n + 148)*x^3 + (288*n^2 - 336*n + 144)*x^4 + (144*n - 144)*x^5)*P[n - 1, x] - ((8*n^2 + 2*n)*x + (76*n^2 + 20*n - 3)*x^2 + (192*n^2 + 80*n - 28)*x^3 + (144*n^2 + 240*n - 56)*x^4 + (288*n + 48)*x^5 + 144*x^6)*D[P[n - 1, x], x])/(16*n^3 - 20*n^2 + 6*n + (24*n^3 - 40*n^2 + 10*n)*x + (48*n^2 - 80*n)*x^2 + 24*n*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^3 - 4*n^2 - 2*n + (120*n^3 - 92*n^2 - 50*n)*x + (240*n^3 - 264*n^2 - 192*n + 32)*x^2 + (144*n^3 + 48*n^2 - 348*n + 148)*x^3 + (288*n^2 - 336*n + 144)*x^4 + (144*n - 144)*x^5)*v[i - 1] - ((8*n^2 + 2*n)*x + (76*n^2 + 20*n - 3)*x^2 + (192*n^2 + 80*n - 28)*x^3 + (144*n^2 + 240*n - 56)*x^4 + (288*n + 48)*x^5 + 144*x^6)*deriv(v[i - 1], x))/(16*n^3 - 20*n^2 + 6*n + (24*n^3 - 40*n^2 + 10*n)*x + (48*n^2 - 80*n)*x^2 + 24*n*x^3)); 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-30cb3cad216ed189","steps":["reverse","D+1"],"edges":["e-73dd335e950d1f81","e-3d49c7c00a50828f"],"text":"A207823 \u2192 reverse \u2192 D+1"},"headline":"A207823 \u2192 reverse \u2192 D+1","chain":null}