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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":"4388b1faa73295320ce0be98ac8f83127b13e9ab5314edea112445c493301c4e","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:  1;\n  1:  4,   3;\n  2:  7,  16,    5;\n  3: 10,  39,   36,    7;\n  4: 13,  72,  114,   64,    9;\n  5: 16, 115,  260,  250,  100,   11;\n  6: 19, 168,  495,  680,  465,  144,   13;\n  7: 22, 231,  840, 1505, 1470,  777,  196,   15;\n  8: 25, 304, 1316, 2912, 3710, 2800, 1204,  256,  17;\n  9: 28, 387, 1944, 5124, 8064, 7938, 4872, 1764, 324, 19;\n","recurrence_sentence":"The row polynomials satisfy (3*n-2+(2*n^2-n+1)*x+(2*n-1)*x^2)*P(n,x) = (3*n+1+(2*n^2+6*n+3)*x+(2*n^2+5*n+3)*x^2+(2*n+1)*x^3)*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[((3*n + 1 + (2*n^2 + 6*n + 3)*x + (2*n^2 + 5*n + 3)*x^2 + (2*n + 1)*x^3)*P[n - 1, x])/(3*n - 2 + (2*n^2 - n + 1)*x + (2*n - 1)*x^2)]];\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] = ((3*n + 1 + (2*n^2 + 6*n + 3)*x + (2*n^2 + 5*n + 3)*x^2 + (2*n + 1)*x^3)*v[i - 1])/(3*n - 2 + (2*n^2 - n + 1)*x + (2*n - 1)*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-c8c19eba9d68881f","steps":["2\u03b8+1","D+1"],"edges":["e-c4ea45619bc7f60a","e-1ff349e71c285f7b"],"text":"A007318 \u2192 2\u03b8+1 \u2192 D+1"},"headline":"A007318 \u2192 2\u03b8+1 \u2192 D+1","chain":null}