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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":"5150ee13ea1b553abc81b69115e36c50b154edf770a82694b335cd5c859305db","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-8409b69bc5d8591b","role":"sequence","name":"basis_map_A013610 of Exponential Riordan array (1, x(1+x/2)).","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:     3;\n  2:     9,     18;\n  3:    27,    108,      81;\n  4:    84,    486,     810,      324;\n  5:   270,   2070,    5265,     4860,     1215;\n  6:   918,   8640,   29565,    43740,    25515,     4374;\n  7:  3234,  36288,  155925,   317520,   306180,   122472,    15309;\n  8: 11952, 154728,  798336,  2075220,  2823660,  1898316,   551124,   52488;\n  9: 45684, 671976, 4048380, 12798324, 22580775, 21861252, 10746918, 2361960, 177147;\n","recurrence_sentence":"The row polynomials satisfy (3*n^2-9*n+6-(18*n-18)*x)*P(n,x) = -(3*n^3-21*n^2+30*n-(27*n^2-81*n)*x+54*n*x^2)*P(n-1,x) + (2*n^2-6*n+(6*n^2-30*n)*x-36*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n^3 - 21*n^2 + 30*n - (27*n^2 - 81*n)*x + 54*n*x^2)*P[n - 1, x] + (2*n^2 - 6*n + (6*n^2 - 30*n)*x - 36*n*x^2)*D[P[n - 1, x], x])/(3*n^2 - 9*n + 6 - (18*n - 18)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n^3 - 21*n^2 + 30*n - (27*n^2 - 81*n)*x + 54*n*x^2)*v[i - 1] + (2*n^2 - 6*n + (6*n^2 - 30*n)*x - 36*n*x^2)*deriv(v[i - 1], x))/(3*n^2 - 9*n + 6 - (18*n - 18)*x)); 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-f6268da8d9636534","steps":["basis A013610","D"],"edges":["e-a15a4571c9a83032","e-2b8043e9364eb23e"],"text":"A122848 \u2192 basis A013610 \u2192 D"},"headline":"A122848 \u2192 basis A013610 \u2192 D","chain":null}