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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":"3f7150e8d423d6bb449f658deeccbe163b73646e16729e0ed8c4eb8b648ab05d","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-1c472d1762258215","role":"sequence","name":"Pascal-(1,8,1) array.","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:  2,    1;\n  2: 11,   12,     1;\n  3: 20,   57,    22,      1;\n  4: 29,  264,   202,     32,      1;\n  5: 38,  633,  1192,    446,     42,      1;\n  6: 47, 1164,  5179,   3776,    789,     52,     1;\n  7: 56, 1857, 14350,  22415,   8988,   1231,    62,    1;\n  8: 65, 2712, 30892,  95000,  70566,  17800,  1772,   72,  1;\n  9: 74, 3729, 56992, 286412, 402036, 177334, 31184, 2412, 82, 1;\n","recurrence_sentence":"The row polynomials satisfy -(9*n^3-25*n^2+16*n-(9*n^3-40*n^2+82*n)*x-(32*n^2-79*n)*x^2-(n^2+30*n)*x^3-n*x^4)*P(n,x) = -(9*n^3-16*n^2+7*n+(153*n^3-554*n^2+299*n)*x+(535*n^2-1341*n+849)*x^2+(17*n^2+479*n-511)*x^3+(16*n-49)*x^4-x^5)*P(n-1,x) + ((9*n^2-7*n)*x+(306*n^2-205*n-16)*x^2+(9*n^2+1116*n-511)*x^3+(67*n+1107)*x^4+(n+67)*x^5+x^6)*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[(-(9*n^3 - 16*n^2 + 7*n + (153*n^3 - 554*n^2 + 299*n)*x + (535*n^2 - 1341*n + 849)*x^2 + (17*n^2 + 479*n - 511)*x^3 + (16*n - 49)*x^4 - x^5)*P[n - 1, x] + ((9*n^2 - 7*n)*x + (306*n^2 - 205*n - 16)*x^2 + (9*n^2 + 1116*n - 511)*x^3 + (67*n + 1107)*x^4 + (n + 67)*x^5 + x^6)*D[P[n - 1, x], x])/(-9*n^3 + 25*n^2 - 16*n + (9*n^3 - 40*n^2 + 82*n)*x + (32*n^2 - 79*n)*x^2 + (n^2 + 30*n)*x^3 + n*x^4)]];\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] = (-(9*n^3 - 16*n^2 + 7*n + (153*n^3 - 554*n^2 + 299*n)*x + (535*n^2 - 1341*n + 849)*x^2 + (17*n^2 + 479*n - 511)*x^3 + (16*n - 49)*x^4 - x^5)*v[i - 1] + ((9*n^2 - 7*n)*x + (306*n^2 - 205*n - 16)*x^2 + (9*n^2 + 1116*n - 511)*x^3 + (67*n + 1107)*x^4 + (n + 67)*x^5 + x^6)*deriv(v[i - 1], x))/(-9*n^3 + 25*n^2 - 16*n + (9*n^3 - 40*n^2 + 82*n)*x + (32*n^2 - 79*n)*x^2 + (n^2 + 30*n)*x^3 + n*x^4)); 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-1c472d1762258215","steps":["D+1"],"edges":["e-34544b66b04cbaa0"],"text":"A143683 \u2192 D+1"},"headline":"A143683 \u2192 D+1","chain":null}