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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":"9b0a5057a440528dbc962168ef5f9701a2110a1bd0e844dc89255828e728ced2","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-0836bf76d9ba1ba5","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) = binomial(k,n-k).","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,     2;\n  2:   11,    16,     4;\n  3:   25,    68,    48,      8;\n  4:   52,   212,   276,    128,     16;\n  5:  103,   562,  1100,    928,    320,     32;\n  6:  197,  1352,  3564,   4512,   2800,    768,     64;\n  7:  368,  3046, 10140,  17328,  16080,   7872,   1792,   128;\n  8:  675,  6548, 26388,  57088,  71520,  52160,  21056,  4096,  256;\n  9: 1221, 13588, 64356, 168944, 268272, 263904, 158144, 54272, 9216, 512;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-12*n^2-34*n-14+(16*n^2-52*n-68)*x+(16*n^2+8*n-8)*x^2+(16*n+16)*x^3)*P(n,x) = -(16*n^3-136*n^2+252*n+76-(32*n^2-8*n-544)*x+(32*n^2-112*n+224)*x^2-(32*n+128)*x^3-64*x^4)*P(n-1,x) + (20*n^2-50*n-15+(48*n^2-80*n-126)*x+(16*n^2+96*n-168)*x^2+(128*n+112)*x^3+(32*n+144)*x^4+32*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[(-(16*n^3 - 136*n^2 + 252*n + 76 - (32*n^2 - 8*n - 544)*x + (32*n^2 - 112*n + 224)*x^2 - (32*n + 128)*x^3 - 64*x^4)*P[n - 1, x] + (20*n^2 - 50*n - 15 + (48*n^2 - 80*n - 126)*x + (16*n^2 + 96*n - 168)*x^2 + (128*n + 112)*x^3 + (32*n + 144)*x^4 + 32*x^5)*D[P[n - 1, x], x])/(8*n^3 - 12*n^2 - 34*n - 14 + (16*n^2 - 52*n - 68)*x + (16*n^2 + 8*n - 8)*x^2 + (16*n + 16)*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 - 136*n^2 + 252*n + 76 - (32*n^2 - 8*n - 544)*x + (32*n^2 - 112*n + 224)*x^2 - (32*n + 128)*x^3 - 64*x^4)*v[i - 1] + (20*n^2 - 50*n - 15 + (48*n^2 - 80*n - 126)*x + (16*n^2 + 96*n - 168)*x^2 + (128*n + 112)*x^3 + (32*n + 144)*x^4 + 32*x^5)*deriv(v[i - 1], x))/(8*n^3 - 12*n^2 - 34*n - 14 + (16*n^2 - 52*n - 68)*x + (16*n^2 + 8*n - 8)*x^2 + (16*n + 16)*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-21854f4257effb01","steps":["basis A013609","D+1"],"edges":["e-38aa524f6e49a0d8","e-51a8a8033f224ded"],"text":"A030528 \u2192 basis A013609 \u2192 D+1"},"headline":"A030528 \u2192 basis A013609 \u2192 D+1","chain":null}