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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":"c5ff8c5d639dcfb950c810cb3eac2a05722452403cf859f48ffccdb2456d4e9d","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:      1;\n  1:      3,      1;\n  2:     10,      7,      1;\n  3:     35,     39,     12,      1;\n  4:    129,    200,     99,     18,      1;\n  5:    497,    995,    695,    205,     25,      1;\n  6:   1996,   4917,   4485,   1885,    375,     33,     1;\n  7:   8325,  24409,  27657,  15330,   4375,    630,    42,    1;\n  8:  35965, 122488, 166474, 115612,  43855,   9100,   994,   52,  1;\n  9: 160523, 623385, 990558, 831306, 398097, 110439, 17430, 1494, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n+2+(n^2+3*n+2)*x+2*n*x^2+2*x^3)*P(n,x) = -(n^3-7*n-6+(n^3-15*n-8)*x+(n^2-7*n-6)*x^2-8*x^3-2*x^4)*P(n-1,x) + (2*n^2+6*n+4+(4*n^2+16*n+12)*x+(2*n^2+14*n+16)*x^2+(4*n+12)*x^3+4*x^4)*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[(-(n^3 - 7*n - 6 + (n^3 - 15*n - 8)*x + (n^2 - 7*n - 6)*x^2 - 8*x^3 - 2*x^4)*P[n - 1, x] + (2*n^2 + 6*n + 4 + (4*n^2 + 16*n + 12)*x + (2*n^2 + 14*n + 16)*x^2 + (4*n + 12)*x^3 + 4*x^4)*D[P[n - 1, x], x])/(2*n + 2 + (n^2 + 3*n + 2)*x + 2*n*x^2 + 2*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] = (-(n^3 - 7*n - 6 + (n^3 - 15*n - 8)*x + (n^2 - 7*n - 6)*x^2 - 8*x^3 - 2*x^4)*v[i - 1] + (2*n^2 + 6*n + 4 + (4*n^2 + 16*n + 12)*x + (2*n^2 + 14*n + 16)*x^2 + (4*n + 12)*x^3 + 4*x^4)*deriv(v[i - 1], x))/(2*n + 2 + (n^2 + 3*n + 2)*x + 2*n*x^2 + 2*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-07e5356ad37f61ea","steps":["x \u21a6 x+1","D+1"],"edges":["e-2fc15f707287930e","e-210be0f4102dc278"],"text":"A100862 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A100862 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}