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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":"ca330eef61e5d5150c9b3cb20a99e9cd448d93048b6b85884969ad67dfb5cec4","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-7b6a59413dc1b477","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.","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:    0;\n  2:    1;\n  3:    5,      2;\n  4:   18,     30,       3;\n  5:   58,    258,     111,       4;\n  6:  179,   1754,    1938,     332,       5;\n  7:  543,  10560,   24090,   10740,     885,      6;\n  8: 1636,  59316,  247830,  225140,   50010,   2202,    7;\n  9: 4916, 319484, 2268654, 3678460, 1675810, 208662, 5243, 8;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(3*n-3+(2*n^2-8*n+6)*x+(n^3-6*n^2+11*n-6)*x^2)*P(n-2,x) - ((2*n-2)*x+(2*n^2-7*n+5)*x^2-(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (4*n-7+(2*n^2-8*n+8)*x)*P(n-1,x) + ((2*n-4)*x-(n-2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 3 + (2*n^2 - 8*n + 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*P[n - 2, x] - ((2*n - 2)*x + (2*n^2 - 7*n + 5)*x^2 - (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (4*n - 7 + (2*n^2 - 8*n + 8)*x)*P[n - 1, x] + ((2*n - 4)*x - (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = (-(3*n - 3 + (2*n^2 - 8*n + 6)*x + (n^3 - 6*n^2 + 11*n - 6)*x^2)*v[i - 2] - ((2*n - 2)*x + (2*n^2 - 7*n + 5)*x^2 - (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (4*n - 7 + (2*n^2 - 8*n + 8)*x)*v[i - 1] + ((2*n - 4)*x - (n - 2)*x^2)*deriv(v[i - 1], x))/(n - 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-7b6a59413dc1b477","steps":["D"],"edges":["e-2922bfe17192604c"],"text":"A102365 \u2192 D"},"headline":"A102365 \u2192 D","chain":null}