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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":"e30917a7283ac4457e2d1e9a032f19e84ef49c48d10bf94e6560f709d5ee12b5","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-d4c5acf24bc40277","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.","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:   3;\n  2:   6,   1;\n  3:  12,   5;\n  4:  23,  16,   1;\n  5:  41,  44,   7;\n  6:  68, 109,  30,   1;\n  7: 106, 245, 104,   9;\n  8: 157, 504, 314,  48,  1;\n  9: 223, 960, 846, 200, 11;\n","recurrence_sentence":"The row polynomials satisfy -(4*n^3-18*n^2+50*n-24-(n^3-18*n^2+81*n-32)*x-(4*n^2-26*n-6)*x^2-(4*n+4)*x^3)*P(n,x) = -(4*n^3-6*n^2+26*n+12+(3*n^3-8*n^2-21*n-70)*x-(n^3-16*n^2+63*n-132)*x^2-(4*n^2-26*n+46)*x^3-(4*n-4)*x^4)*P(n-1,x) - ((8*n^2-16*n+24)*x-(10*n^2-50*n+100)*x^2+(2*n^2-42*n+120)*x^3+(8*n-52)*x^4+8*x^5)*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[(-(4*n^3 - 6*n^2 + 26*n + 12 + (3*n^3 - 8*n^2 - 21*n - 70)*x - (n^3 - 16*n^2 + 63*n - 132)*x^2 - (4*n^2 - 26*n + 46)*x^3 - (4*n - 4)*x^4)*P[n - 1, x] - ((8*n^2 - 16*n + 24)*x - (10*n^2 - 50*n + 100)*x^2 + (2*n^2 - 42*n + 120)*x^3 + (8*n - 52)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(-4*n^3 + 18*n^2 - 50*n + 24 + (n^3 - 18*n^2 + 81*n - 32)*x + (4*n^2 - 26*n - 6)*x^2 + (4*n + 4)*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] = (-(4*n^3 - 6*n^2 + 26*n + 12 + (3*n^3 - 8*n^2 - 21*n - 70)*x - (n^3 - 16*n^2 + 63*n - 132)*x^2 - (4*n^2 - 26*n + 46)*x^3 - (4*n - 4)*x^4)*v[i - 1] - ((8*n^2 - 16*n + 24)*x - (10*n^2 - 50*n + 100)*x^2 + (2*n^2 - 42*n + 120)*x^3 + (8*n - 52)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(-4*n^3 + 18*n^2 - 50*n + 24 + (n^3 - 18*n^2 + 81*n - 32)*x + (4*n^2 - 26*n - 6)*x^2 + (4*n + 4)*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-d4c5acf24bc40277","steps":["D+1"],"edges":["e-9109e15e9cb5c320"],"text":"A208509 \u2192 D+1"},"headline":"A208509 \u2192 D+1","chain":null}