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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":"15d92fbfb4896ef1f0a84e1bf296e7af9a935689b6a49bec87936e166cf3d1f4","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-6202e291a2627bce","role":"sequence","name":"Triangle, row sums = number of edges in n-dimensional hypercubes.","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:  3;\n  2:  6,  10;\n  3:  9,  30,   21;\n  4: 12,  60,   84,   36;\n  5: 15, 100,  210,  180,   55;\n  6: 18, 150,  420,  540,  330,   78;\n  7: 21, 210,  735, 1260, 1155,  546,  105;\n  8: 24, 280, 1176, 2520, 3080, 2184,  840,  136;\n  9: 27, 360, 1764, 4536, 6930, 6552, 3780, 1224, 171;\n","recurrence_sentence":"The row polynomials satisfy (3*n-3+(2*n^2-3*n+1)*x)*P(n,x) = (3*n+(2*n^2+4*n)*x+(2*n^2+n)*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n + (2*n^2 + 4*n)*x + (2*n^2 + n)*x^2)*P[n - 1, x])/(3*n - 3 + (2*n^2 - 3*n + 1)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((3*n + (2*n^2 + 4*n)*x + (2*n^2 + n)*x^2)*v[i - 1])/(3*n - 3 + (2*n^2 - 3*n + 1)*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-c8c19eba9d68881f","steps":["2\u03b8+1","D"],"edges":["e-c4ea45619bc7f60a","e-ef8d93833f370225"],"text":"A007318 \u2192 2\u03b8+1 \u2192 D"},"headline":"A007318 \u2192 2\u03b8+1 \u2192 D","chain":null}