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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":"51820da12bc8a9e398b978f44b0a2cba4dab0117fab216a2a493b0202b2482f5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-c1c65e1c83bcd2f2","role":"sequence","name":"Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  0;\n   2: -1;\n   3: -1,  -2;\n   4: -1,  -4,  3;\n   5: -1,  -6,  6,  4;\n   6: -1,  -8,  9, 12,  -5;\n   7: -1, -10, 12, 24, -15,   -6;\n   8: -1, -12, 15, 40, -30,  -24,  7;\n   9: -1, -14, 18, 60, -50,  -60, 28,  8;\n  10: -1, -16, 21, 84, -75, -120, 70, 40, -9;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-14*n+12+(8*n^3-52*n^2+108*n-72)*x)*P(n,x) = (4*n^2-14*n+12+(8*n^3-40*n^2+62*n-30)*x-(12*n^2-52*n+48)*x^2-(16*n^3-96*n^2+176*n-96)*x^3)*P(n-1,x) - ((2*n-3)*x+(4*n^2-12*n+8)*x^2-(8*n-12)*x^3-(16*n^2-48*n+32)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = -1.","mathematica":"P[1, x_] := 0;\nP[2, x_] := -1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 14*n + 12 + (8*n^3 - 40*n^2 + 62*n - 30)*x - (12*n^2 - 52*n + 48)*x^2 - (16*n^3 - 96*n^2 + 176*n - 96)*x^3)*P[n - 1, x] - ((2*n - 3)*x + (4*n^2 - 12*n + 8)*x^2 - (8*n - 12)*x^3 - (16*n^2 - 48*n + 32)*x^4)*D[P[n - 1, x], x])/(4*n^2 - 14*n + 12 + (8*n^3 - 52*n^2 + 108*n - 72)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = -1; for(n = 3, N, my(i = n); v[i] = ((4*n^2 - 14*n + 12 + (8*n^3 - 40*n^2 + 62*n - 30)*x - (12*n^2 - 52*n + 48)*x^2 - (16*n^3 - 96*n^2 + 176*n - 96)*x^3)*v[i - 1] - ((2*n - 3)*x + (4*n^2 - 12*n + 8)*x^2 - (8*n - 12)*x^3 - (16*n^2 - 48*n + 32)*x^4)*deriv(v[i - 1], x))/(4*n^2 - 14*n + 12 + (8*n^3 - 52*n^2 + 108*n - 72)*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c1c65e1c83bcd2f2","steps":["D"],"edges":["e-df27c1125e074e5e"],"text":"A108299 \u2192 D"},"headline":"A108299 \u2192 D","chain":null}