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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":"6ead0a9d9f408f700f4ed21e983c56b91be2325e583babb430024a820969b56e","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-216193c52744946f","role":"sequence","name":"basis_map_A007318 of Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:     7,      3;\n  2:    22,     18,      3;\n  3:    76,     96,     33,      3;\n  4:   244,    420,    234,     48,      3;\n  5:   772,   1692,   1320,    444,     63,     3;\n  6:  2392,   6384,   6456,   3096,    726,    78,     3;\n  7:  7312,  23040,  28728,  18072,   6084,  1080,    93,    3;\n  8: 22096,  80352, 119376,  93456,  41580, 10620,  1506,  108,   3;\n  9: 66160, 272880, 471168, 442656, 246636, 83556, 17040, 2004, 123, 3;\n","recurrence_sentence":"The row polynomials satisfy -(12*n-(3*n^3+12*n^2-21*n-24)*x-(12*n^2+2*n-36)*x^2-(n^2+10*n-13)*x^3-(n-1)*x^4)*P(n,x) = -(18*n^3+54*n^2+12*n+(15*n^3+105*n^2+102*n+24)*x+(60*n^2+148*n+36)*x^2+(5*n^2+62*n+13)*x^3+(5*n+1)*x^4)*P(n-1,x) + (36*n^2+180*n+216+(72*n^2+504*n+780)*x+(39*n^2+495*n+1086)*x^2+(3*n^2+195*n+719)*x^3+(25*n+222)*x^4+(n+26)*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[(-(18*n^3 + 54*n^2 + 12*n + (15*n^3 + 105*n^2 + 102*n + 24)*x + (60*n^2 + 148*n + 36)*x^2 + (5*n^2 + 62*n + 13)*x^3 + (5*n + 1)*x^4)*P[n - 1, x] + (36*n^2 + 180*n + 216 + (72*n^2 + 504*n + 780)*x + (39*n^2 + 495*n + 1086)*x^2 + (3*n^2 + 195*n + 719)*x^3 + (25*n + 222)*x^4 + (n + 26)*x^5 + x^6)*D[P[n - 1, x], x])/(-12*n + (3*n^3 + 12*n^2 - 21*n - 24)*x + (12*n^2 + 2*n - 36)*x^2 + (n^2 + 10*n - 13)*x^3 + (n - 1)*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] = (-(18*n^3 + 54*n^2 + 12*n + (15*n^3 + 105*n^2 + 102*n + 24)*x + (60*n^2 + 148*n + 36)*x^2 + (5*n^2 + 62*n + 13)*x^3 + (5*n + 1)*x^4)*v[i - 1] + (36*n^2 + 180*n + 216 + (72*n^2 + 504*n + 780)*x + (39*n^2 + 495*n + 1086)*x^2 + (3*n^2 + 195*n + 719)*x^3 + (25*n + 222)*x^4 + (n + 26)*x^5 + x^6)*deriv(v[i - 1], x))/(-12*n + (3*n^3 + 12*n^2 - 21*n - 24)*x + (12*n^2 + 2*n - 36)*x^2 + (n^2 + 10*n - 13)*x^3 + (n - 1)*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-eac80c2bf8e566f5","steps":["x \u21a6 x+1","D+1"],"edges":["e-237035d3e801d2dc","e-d949b5de47836f37"],"text":"A249250 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A249250 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}