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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":"3f349c595fcb64640f1a8be7be9564d66e03e92102e6780a41b6366eae1c66cd","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-107a00b591d3c4c8","role":"sequence","name":"basis_map_A013609 of Triangular array formed by the Mersenne numbers.","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:      6,       2;\n  2:     30,      24,       4;\n  3:    136,     184,      72,       8;\n  4:    580,    1136,     744,     192,      16;\n  5:   2376,    6168,    5824,    2496,     480,      32;\n  6:   9464,   30784,   38496,   23680,    7520,    1152,     64;\n  7:  36928,  144832,  227136,  184640,   83840,   21120,   2688,    128;\n  8: 141840,  652416, 1235648, 1258752,  753600,  270592,  56448,   6144,   256;\n  9: 538080, 2842784, 6326016, 7783168, 5832064, 2755200, 817152, 145408, 13824, 512;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n-n*x-n*x^2)*P(n,x) = (2*n^3+6*n^2+10*n+6+(n^3+3*n^2-2)*x-(4*n+4)*x^2-(n+1)*x^3)*P(n-1,x) + (2*n^2+2*n+(4*n^2+4*n-6)*x+(n^2+n-14)*x^2-7*x^3-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[((2*n^3 + 6*n^2 + 10*n + 6 + (n^3 + 3*n^2 - 2)*x - (4*n + 4)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (2*n^2 + 2*n + (4*n^2 + 4*n - 6)*x + (n^2 + n - 14)*x^2 - 7*x^3 - x^4)*D[P[n - 1, x], x])/(n^3 + 3*n - n*x - n*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] = ((2*n^3 + 6*n^2 + 10*n + 6 + (n^3 + 3*n^2 - 2)*x - (4*n + 4)*x^2 - (n + 1)*x^3)*v[i - 1] + (2*n^2 + 2*n + (4*n^2 + 4*n - 6)*x + (n^2 + n - 14)*x^2 - 7*x^3 - x^4)*deriv(v[i - 1], x))/(n^3 + 3*n - n*x - n*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-9b5c7e97e5191da6","steps":["basis A013609","D+1"],"edges":["e-3133312db52cd60e","e-bd4e311489df1347"],"text":"A110441 \u2192 basis A013609 \u2192 D+1"},"headline":"A110441 \u2192 basis A013609 \u2192 D+1","chain":null}