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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":"5bed49eaa34c7c5b055592f4fc5befbded879a1cd9dc905db39064253c287014","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-131f5baf2477adf2","role":"sequence","name":"basis_map_A007318 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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:      0;\n  1:      1;\n  2:      8,      2;\n  3:     42,     24,      3;\n  4:    190,    174,     48,      4;\n  5:    797,   1010,    444,     80,      5;\n  6:   3192,   5178,   3156,    900,    120,     6;\n  7:  12391,  24522,  19194,   7580,   1590,   168,    7;\n  8:  47030, 109910, 105288,  53440,  15490,  2562,  224,   8;\n  9: 175494, 473094, 536706, 333876, 124335, 28350, 3864, 288, 9;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^3-28*n^2+57*n-32-(3*n^3-2*n^2-15*n+14)*x+(2*n^2-2)*x^2)*P(n,x) = -(24*n^3-152*n^2+210*n-52+(12*n^3-128*n^2+222*n-40)*x-(20*n^2-66*n+4)*x^2+6*n*x^3)*P(n-1,x) + (39*n^2-91*n+26+(69*n^2-187*n+46)*x+(33*n^2-123*n+22)*x^2+(3*n^2-29*n+2)*x^3-2*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(24*n^3 - 152*n^2 + 210*n - 52 + (12*n^3 - 128*n^2 + 222*n - 40)*x - (20*n^2 - 66*n + 4)*x^2 + 6*n*x^3)*P[n - 1, x] + (39*n^2 - 91*n + 26 + (69*n^2 - 187*n + 46)*x + (33*n^2 - 123*n + 22)*x^2 + (3*n^2 - 29*n + 2)*x^3 - 2*n*x^4)*D[P[n - 1, x], x])/(-3*n^3 + 28*n^2 - 57*n + 32 + (3*n^3 - 2*n^2 - 15*n + 14)*x - (2*n^2 - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(24*n^3 - 152*n^2 + 210*n - 52 + (12*n^3 - 128*n^2 + 222*n - 40)*x - (20*n^2 - 66*n + 4)*x^2 + 6*n*x^3)*v[i - 1] + (39*n^2 - 91*n + 26 + (69*n^2 - 187*n + 46)*x + (33*n^2 - 123*n + 22)*x^2 + (3*n^2 - 29*n + 2)*x^3 - 2*n*x^4)*deriv(v[i - 1], x))/(-3*n^3 + 28*n^2 - 57*n + 32 + (3*n^3 - 2*n^2 - 15*n + 14)*x - (2*n^2 - 2)*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-d6b7687ba0ff59dd","steps":["basis A007318","D"],"edges":["e-32d1efa81dadfdb2","e-d0d9dc85b0ed2bec"],"text":"A116412 \u2192 basis A007318 \u2192 D"},"headline":"A116412 \u2192 basis A007318 \u2192 D","chain":null}