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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":"05ed106c0bd8d5b8217fc446fcd29ed289a4d5cada96b57eaf8a56e1a9ccabc6","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-632227a1ddc9eaa6","role":"sequence","name":"Binomial coefficients C(n,k) with n-k odd, read by rows.","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:   2;\n  2:   4,   3;\n  3:   8,   4;\n  4:  11,  20,   5;\n  5:  26,  32,   6;\n  6:  22,  91,  56,   7;\n  7:  64, 168,  80,   8;\n  8:  37, 288, 378, 120,  9;\n  9: 130, 624, 612, 160, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9-15*x+3*x^2+3*x^3)*P(n-6,x) + (4-12*x+12*x^2-4*x^3)*d/dx P(n-6,x) - (31-10*x-5*x^2)*P(n-4,x) - (8-8*x^2)*d/dx P(n-4,x) + (9-x)*P(n-2,x) + (4-4*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 4 + 3*x, P(3,x) = 8 + 4*x, P(4,x) = 11 + 20*x + 5*x^2, P(5,x) = 26 + 32*x + 6*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 4 + 3*x;\nP[3, x_] := 8 + 4*x;\nP[4, x_] := 11 + 20*x + 5*x^2;\nP[5, x_] := 26 + 32*x + 6*x^2;\nP[n_, x_] := P[n, x] = Expand[-(9 - 15*x + 3*x^2 + 3*x^3)*P[n - 6, x] + (4 - 12*x + 12*x^2 - 4*x^3)*D[P[n - 6, x], x] - (31 - 10*x - 5*x^2)*P[n - 4, x] - (8 - 8*x^2)*D[P[n - 4, x], x] + (9 - x)*P[n - 2, x] + (4 - 4*x)*D[P[n - 2, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 4 + 3*x; v[4] = 8 + 4*x; v[5] = 11 + 20*x + 5*x^2; v[6] = 26 + 32*x + 6*x^2; for(n = 6, N, my(i = n + 1); v[i] = -(9 - 15*x + 3*x^2 + 3*x^3)*v[i - 6] + (4 - 12*x + 12*x^2 - 4*x^3)*deriv(v[i - 6], x) - (31 - 10*x - 5*x^2)*v[i - 4] - (8 - 8*x^2)*deriv(v[i - 4], x) + (9 - x)*v[i - 2] + (4 - 4*x)*deriv(v[i - 2], x)); 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-e655e253465c7aff","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-c2ab5705bc1e09a1","e-42651b87961ddc20"],"text":"Veronese parent of A109447 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"Veronese parent of A109447 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}