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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":"ea6dab962623576658803dcdd2d88e5c68d77c62ad511ad134dca19fd8db68d6","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-6bbd968fe5cbbe11","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'.","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:      0;\n  1:      1;\n  2:      5,      2;\n  3:     21,     18,      3;\n  4:     86,    114,     42,      4;\n  5:    355,    640,    375,     80,      5;\n  6:   1497,   3420,   2775,    960,    135,     6;\n  7:   6475,  17934,  18690,   9100,   2100,   210,    7;\n  8:  28772,  93716, 119616,  75740,  24920,  4116,  308,   8;\n  9: 131337, 492048, 744534, 583128, 252315, 59976, 7434, 432, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2+(n^2-n)*x-(2*n-2)*x^2)*P(n,x) = -(n^3-4*n^2-n+(n^3-8*n^2+5*n)*x-(3*n^2-9*n)*x^2+2*n*x^3)*P(n-1,x) + (2*n^2-2*n+(4*n^2-8*n)*x+(2*n^2-10*n)*x^2-4*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 - n + (n^3 - 8*n^2 + 5*n)*x - (3*n^2 - 9*n)*x^2 + 2*n*x^3)*P[n - 1, x] + (2*n^2 - 2*n + (4*n^2 - 8*n)*x + (2*n^2 - 10*n)*x^2 - 4*n*x^3)*D[P[n - 1, x], x])/(2*n - 2 + (n^2 - n)*x - (2*n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 - n + (n^3 - 8*n^2 + 5*n)*x - (3*n^2 - 9*n)*x^2 + 2*n*x^3)*v[i - 1] + (2*n^2 - 2*n + (4*n^2 - 8*n)*x + (2*n^2 - 10*n)*x^2 - 4*n*x^3)*deriv(v[i - 1], x))/(2*n - 2 + (n^2 - n)*x - (2*n - 2)*x^2)); 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-07e5356ad37f61ea","steps":["x \u21a6 x+1","D"],"edges":["e-2fc15f707287930e","e-c61bf91516d9ab55"],"text":"A100862 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A100862 \u2192 x \u21a6 x+1 \u2192 D","chain":null}