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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":"8a2bcd028f7a9341721bd9f1d89d5f36b08d6a22a07d24484175bb9891d68090","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-fac4d4a000d744e8","role":"sequence","name":"Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:     5,     2;\n  3:    20,    18,      3;\n  4:    71,   102,     42,      4;\n  5:   235,   466,    315,     80,     5;\n  6:   744,  1884,   1782,    760,   135,     6;\n  7:  2285,  7044,   8580,   5180,  1575,   210,    7;\n  8:  6865, 24946,  37206,  29140, 12670,  2940,  308,   8;\n  9: 20284, 84894, 149889, 144488, 82035, 27468, 5082, 432, 9;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-10*n+6)*P(n,x) = (16*n^2-32*n+10+(12*n^2-26*n+8)*x)*P(n-1,x) - (10*n-5+(18*n-9)*x+(8*n-4)*x^2)*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[((16*n^2 - 32*n + 10 + (12*n^2 - 26*n + 8)*x)*P[n - 1, x] - (10*n - 5 + (18*n - 9)*x + (8*n - 4)*x^2)*D[P[n - 1, x], x])/(4*n^2 - 10*n + 6)]];\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] = ((16*n^2 - 32*n + 10 + (12*n^2 - 26*n + 8)*x)*v[i - 1] - (10*n - 5 + (18*n - 9)*x + (8*n - 4)*x^2)*deriv(v[i - 1], x))/(4*n^2 - 10*n + 6)); 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-fac4d4a000d744e8","steps":["D"],"edges":["e-2dfe8925c3432b87"],"text":"A060920 \u2192 D"},"headline":"A060920 \u2192 D","chain":null}