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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":"b3dae189c4735f75f5307f3b1b546a589d6e40b6a8f7af1bf614ffad0e5ab77d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-dfab35176bd08e72","role":"sequence","name":"Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; 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:  11,   11,     2;\n  1:  21,   38,    17,     2;\n  2:  36,  102,    81,    23,      2;\n  3:  57,  232,   283,   140,     29,      2;\n  4:  85,  469,   808,   604,    215,     35,     2;\n  5: 121,  868,  1998,  2080,   1105,    306,    41,     2;\n  6: 166, 1500,  4434,  6094,   4460,   1826,   413,    47,    2;\n  7: 221, 2454,  9042, 15796,  15119,   8456,  2807,   536,   53,   2;\n  8: 287, 3839, 17226, 37180,  44863,  32571, 14672,  4088,  675,  59,  2;\n  9: 365, 5786, 31031, 80938, 119795, 108920, 63308, 23808, 5709, 830, 65, 2;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^3+18*n^2+52*n+60-(4*n^3+18*n^2+36*n+30)*x+(6*n+6)*x^2+(4*n+6)*x^3)*P(n,x) = -(2*n^3+24*n^2+94*n+132-(3*n^3+14*n^2+21*n+6)*x-(2*n^3+14*n^2+24*n+20)*x^2+(9*n+14)*x^3+(2*n+4)*x^4)*P(n-1,x) - ((4*n^2+28*n+48)*x-(7*n^2+25*n+16)*x^2-(2*n^2+8*n-21)*x^3+15*x^4+2*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 11 + 11*x + 2*x^2.","mathematica":"P[0, x_] := 11 + 11*x + 2*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 + 24*n^2 + 94*n + 132 - (3*n^3 + 14*n^2 + 21*n + 6)*x - (2*n^3 + 14*n^2 + 24*n + 20)*x^2 + (9*n + 14)*x^3 + (2*n + 4)*x^4)*P[n - 1, x] - ((4*n^2 + 28*n + 48)*x - (7*n^2 + 25*n + 16)*x^2 - (2*n^2 + 8*n - 21)*x^3 + 15*x^4 + 2*x^5)*D[P[n - 1, x], x])/(-2*n^3 - 18*n^2 - 52*n - 60 + (4*n^3 + 18*n^2 + 36*n + 30)*x - (6*n + 6)*x^2 - (4*n + 6)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 11 + 11*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n^3 + 24*n^2 + 94*n + 132 - (3*n^3 + 14*n^2 + 21*n + 6)*x - (2*n^3 + 14*n^2 + 24*n + 20)*x^2 + (9*n + 14)*x^3 + (2*n + 4)*x^4)*v[i - 1] - ((4*n^2 + 28*n + 48)*x - (7*n^2 + 25*n + 16)*x^2 - (2*n^2 + 8*n - 21)*x^3 + 15*x^4 + 2*x^5)*deriv(v[i - 1], x))/(-2*n^3 - 18*n^2 - 52*n - 60 + (4*n^3 + 18*n^2 + 36*n + 30)*x - (6*n + 6)*x^2 - (4*n + 6)*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-dfab35176bd08e72","steps":["D+1"],"edges":["e-da807e70e77c2903"],"text":"A207606 \u2192 D+1"},"headline":"A207606 \u2192 D+1","chain":null}