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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":"1b47c697985c69f0b4d4dfd59ff85eed9f2bf1f2f33385c29e36c4ed63bebef5","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-89353a17ee1d740a","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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:    4;\n  3:   22,    36;\n  4:   70,   300,    288;\n  5:  170,  1350,   3288,    2400;\n  6:  350,  4410,  19488,   35280,    21600;\n  7:  644, 11760,  81228,  262640,   392040,   211680;\n  8: 1092, 27216, 269388, 1345680,  3543720,  4602528,  2257920;\n  9: 1740, 56700, 759276, 5386500, 21710400, 49253400, 57488256, 26127360;\n","recurrence_sentence":"The row polynomials satisfy (n^3-5*n^2+8*n-4)*P(n,x) = (n^3-2*n^2+n+(3*n^4-12*n^3+15*n^2-6*n)*x+(3*n^5-18*n^4+39*n^3-36*n^2+12*n)*x^2+(n^6-8*n^5+25*n^4-38*n^3+28*n^2-8*n)*x^3)*P(n-3,x) - (3*n^3-9*n^2+7*n+(6*n^4-27*n^3+39*n^2-18*n)*x+(3*n^5-18*n^4+39*n^3-36*n^2+12*n)*x^2)*P(n-2,x) + (3*n^3-12*n^2+14*n-4+(3*n^4-15*n^3+24*n^2-12*n)*x)*P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - 2*n^2 + n + (3*n^4 - 12*n^3 + 15*n^2 - 6*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2 + (n^6 - 8*n^5 + 25*n^4 - 38*n^3 + 28*n^2 - 8*n)*x^3)*P[n - 3, x] - (3*n^3 - 9*n^2 + 7*n + (6*n^4 - 27*n^3 + 39*n^2 - 18*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2)*P[n - 2, x] + (3*n^3 - 12*n^2 + 14*n - 4 + (3*n^4 - 15*n^3 + 24*n^2 - 12*n)*x)*P[n - 1, x])/(n^3 - 5*n^2 + 8*n - 4)]];\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] = 4; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - 2*n^2 + n + (3*n^4 - 12*n^3 + 15*n^2 - 6*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2 + (n^6 - 8*n^5 + 25*n^4 - 38*n^3 + 28*n^2 - 8*n)*x^3)*v[i - 3] - (3*n^3 - 9*n^2 + 7*n + (6*n^4 - 27*n^3 + 39*n^2 - 18*n)*x + (3*n^5 - 18*n^4 + 39*n^3 - 36*n^2 + 12*n)*x^2)*v[i - 2] + (3*n^3 - 12*n^2 + 14*n - 4 + (3*n^4 - 15*n^3 + 24*n^2 - 12*n)*x)*v[i - 1])/(n^3 - 5*n^2 + 8*n - 4)); 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-4d60c3dd8627c01d","steps":["D","D"],"edges":["e-74e2b15799b20e69","e-077fcc58a65967e8"],"text":"A094638 \u2192 D \u2192 D"},"headline":"A094638 \u2192 D \u2192 D","chain":null}