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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":"231f344ee0e6c36ef14ba3fe616ab25dd1d0a27732a0a6f0552d7df85a87d20f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c2648604ab227c74","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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:     5,      2;\n  2:    17,     13,      2;\n  3:    54,     63,     21,      2;\n  4:   166,    263,    139,     29,      2;\n  5:   499,   1003,    735,    245,     37,     2;\n  6:  1475,   3600,   3393,   1578,    381,    45,    2;\n  7:  4303,  12370,  14295,   8596,   2900,   547,   53,   2;\n  8: 12421,  41129,  56389,  41761,  18250,  4809,  743,  61,  2;\n  9: 35544, 133275, 211647, 186657, 100656, 34371, 7413, 969, 69, 2;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-15*n^2-7*n+23+(4*n^3-12*n^2-24*n+44)*x-(22*n-25)*x^2-(4*n-4)*x^3)*P(n,x) = (3*n^3-21*n^2-44*n+77+(7*n^3-16*n^2-84*n+159)*x+(2*n^3-2*n^2-64*n+103)*x^2-(19*n-25)*x^3-(2*n-2)*x^4)*P(n-1,x) + (5*n^2-20*n-100+(16*n^2-34*n-280)*x+(13*n^2-16*n-287)*x^2+(2*n^2-2*n-132)*x^3-27*x^4-2*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[((3*n^3 - 21*n^2 - 44*n + 77 + (7*n^3 - 16*n^2 - 84*n + 159)*x + (2*n^3 - 2*n^2 - 64*n + 103)*x^2 - (19*n - 25)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (5*n^2 - 20*n - 100 + (16*n^2 - 34*n - 280)*x + (13*n^2 - 16*n - 287)*x^2 + (2*n^2 - 2*n - 132)*x^3 - 27*x^4 - 2*x^5)*D[P[n - 1, x], x])/(2*n^3 - 15*n^2 - 7*n + 23 + (4*n^3 - 12*n^2 - 24*n + 44)*x - (22*n - 25)*x^2 - (4*n - 4)*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] = ((3*n^3 - 21*n^2 - 44*n + 77 + (7*n^3 - 16*n^2 - 84*n + 159)*x + (2*n^3 - 2*n^2 - 64*n + 103)*x^2 - (19*n - 25)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (5*n^2 - 20*n - 100 + (16*n^2 - 34*n - 280)*x + (13*n^2 - 16*n - 287)*x^2 + (2*n^2 - 2*n - 132)*x^3 - 27*x^4 - 2*x^5)*deriv(v[i - 1], x))/(2*n^3 - 15*n^2 - 7*n + 23 + (4*n^3 - 12*n^2 - 24*n + 44)*x - (22*n - 25)*x^2 - (4*n - 4)*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-353fb5a029e4e669","steps":["basis A007318","D+1"],"edges":["e-06cd9a9d948af774","e-540a0d58194d9c40"],"text":"A207607 \u2192 basis A007318 \u2192 D+1"},"headline":"A207607 \u2192 basis A007318 \u2192 D+1","chain":null}