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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":"015de005927deda598576b75e5da6ff222ce9d84b3e71ffd34759e138c6abf1e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-5b857304d1bafe7a","role":"sequence","name":"Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":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:    4,      6;\n   4:   11,     50,      45;\n   5:   26,    260,     630,      420;\n   6:   57,   1092,    5250,     8820,      4725;\n   7:  120,   4074,   34104,   107100,    138600,     62370;\n   8:  247,  14142,  191415,   990220,   2304225,   2432430,     945945;\n   9:  502,  46872,  977790,  7758520,  28678650,  52972920,   47297250,   16216200;\n  10: 1013, 150656, 4684548, 54437240, 298147850, 852611760, 1305404100, 1010809800, 310134825;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(2*n-2+(2*n^2-4*n+2)*x+(n^3-4*n^2+5*n-2)*x^2)*P(n-2,x) - ((n-1)*x+(n^2-2*n+1)*x^2+(n^2-3*n+2)*x^3)*d/dx P(n-2,x) + (3*n-5+(2*n^2-4*n)*x)*P(n-1,x) + ((n-2)*x+(n-2)*x^2)*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[(-(2*n - 2 + (2*n^2 - 4*n + 2)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)*P[n - 2, x] - ((n - 1)*x + (n^2 - 2*n + 1)*x^2 + (n^2 - 3*n + 2)*x^3)*D[P[n - 2, x], x] + (3*n - 5 + (2*n^2 - 4*n)*x)*P[n - 1, x] + ((n - 2)*x + (n - 2)*x^2)*D[P[n - 1, x], x])/(n - 2)]];\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] = (-(2*n - 2 + (2*n^2 - 4*n + 2)*x + (n^3 - 4*n^2 + 5*n - 2)*x^2)*v[i - 2] - ((n - 1)*x + (n^2 - 2*n + 1)*x^2 + (n^2 - 3*n + 2)*x^3)*deriv(v[i - 2], x) + (3*n - 5 + (2*n^2 - 4*n)*x)*v[i - 1] + ((n - 2)*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5b857304d1bafe7a","steps":["D"],"edges":["e-5db1dbf0216c250f"],"text":"A112493 \u2192 D"},"headline":"A112493 \u2192 D","chain":null}