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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":"ec6d0171f0522ef222ba65d017d4f8ba271b5451fa02463959cc85837a65d0b9","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-b9dfb234567937e9","role":"sequence","name":"basis_map_A007318 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:      7,      2;\n  3:     35,     22,      3;\n  4:    155,    152,     45,      4;\n  5:    642,    858,    399,     76,      5;\n  6:   2550,   4320,   2757,    824,    115,     6;\n  7:   9841,  20202,  16437,   6756,   1475,   162,    7;\n  8:  37189,  89708,  88851,  46684,  14015,  2400,  217,   8;\n  9: 138305, 383386, 447855, 287192, 110320, 25950, 3647, 280, 9;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^2-14*n+11-(3*n^2-n-2)*x+(n-1)*x^2)*P(n,x) = -(24*n^2-79*n+27+(12*n^2-76*n+38)*x-(13*n-17)*x^2+2*x^3)*P(n-1,x) + (39*n-26+(69*n-59)*x+(33*n-45)*x^2+(3*n-13)*x^3-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^2 - 79*n + 27 + (12*n^2 - 76*n + 38)*x - (13*n - 17)*x^2 + 2*x^3)*P[n - 1, x] + (39*n - 26 + (69*n - 59)*x + (33*n - 45)*x^2 + (3*n - 13)*x^3 - x^4)*D[P[n - 1, x], x])/(-3*n^2 + 14*n - 11 + (3*n^2 - n - 2)*x - (n - 1)*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^2 - 79*n + 27 + (12*n^2 - 76*n + 38)*x - (13*n - 17)*x^2 + 2*x^3)*v[i - 1] + (39*n - 26 + (69*n - 59)*x + (33*n - 45)*x^2 + (3*n - 13)*x^3 - x^4)*deriv(v[i - 1], x))/(-3*n^2 + 14*n - 11 + (3*n^2 - n - 2)*x - (n - 1)*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-d5c14b9fec875ecc","steps":["x \u21a6 x+1","D"],"edges":["e-03e3acc3469b904b","e-cbc65111dd4b15aa"],"text":"A121574 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A121574 \u2192 x \u21a6 x+1 \u2192 D","chain":null}