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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":"4963bec3c99b83d8707f1e48b2ab0f76a0f5c8cbbd9d534a10584ca96ddd9f46","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-3c0665ea87e1dc8e","role":"sequence","name":"basis_map_A007318 of A Riordan array of coordination sequences.","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:     4,     1;\n  2:    13,     8,      1;\n  3:    40,    41,     12,      1;\n  4:   117,   172,     84,     16,     1;\n  5:   332,   641,    450,    142,    20,     1;\n  6:   921,  2212,   2049,    928,   215,    24,    1;\n  7:  2512,  7233,   8370,   5007,  1660,   303,   28,   1;\n  8:  6761, 22728,  31648,  23696, 10370,  2700,  406,  32,  1;\n  9: 18004, 69265, 112956, 101980, 55888, 19182, 4102, 524, 36, 1;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-16+(2*n^3+4*n^2+12*n-24)*x+(6*n^2+4*n+4)*x^2+(n^2+6*n+7)*x^3+(n+1)*x^4)*P(n,x) = -(8*n^3-32*n^2-24*n+80+(6*n^3-14*n^2-90*n+130)*x+(14*n^2-62*n-2)*x^2+(3*n^2-2*n-71)*x^3+(n-23)*x^4-2*x^5)*P(n-1,x) + (16*n^2+16*n-16+(32*n^2+80*n-8)*x+(18*n^2+122*n+86)*x^2+(2*n^2+72*n+145)*x^3+(15*n+82)*x^4+(n+16)*x^5+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[(-(8*n^3 - 32*n^2 - 24*n + 80 + (6*n^3 - 14*n^2 - 90*n + 130)*x + (14*n^2 - 62*n - 2)*x^2 + (3*n^2 - 2*n - 71)*x^3 + (n - 23)*x^4 - 2*x^5)*P[n - 1, x] + (16*n^2 + 16*n - 16 + (32*n^2 + 80*n - 8)*x + (18*n^2 + 122*n + 86)*x^2 + (2*n^2 + 72*n + 145)*x^3 + (15*n + 82)*x^4 + (n + 16)*x^5 + x^6)*D[P[n - 1, x], x])/(8*n^2 - 16 + (2*n^3 + 4*n^2 + 12*n - 24)*x + (6*n^2 + 4*n + 4)*x^2 + (n^2 + 6*n + 7)*x^3 + (n + 1)*x^4)]];\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] = (-(8*n^3 - 32*n^2 - 24*n + 80 + (6*n^3 - 14*n^2 - 90*n + 130)*x + (14*n^2 - 62*n - 2)*x^2 + (3*n^2 - 2*n - 71)*x^3 + (n - 23)*x^4 - 2*x^5)*v[i - 1] + (16*n^2 + 16*n - 16 + (32*n^2 + 80*n - 8)*x + (18*n^2 + 122*n + 86)*x^2 + (2*n^2 + 72*n + 145)*x^3 + (15*n + 82)*x^4 + (n + 16)*x^5 + x^6)*deriv(v[i - 1], x))/(8*n^2 - 16 + (2*n^3 + 4*n^2 + 12*n - 24)*x + (6*n^2 + 4*n + 4)*x^2 + (n^2 + 6*n + 7)*x^3 + (n + 1)*x^4)); 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-ec35d1145e9a1211","steps":["x \u21a6 x+1","D+1"],"edges":["e-0b1886c9648f8656","e-f1045d8d338e852d"],"text":"A113413 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A113413 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}