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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":"50a7f5ef7fae67e20ed012648ea931a793efbd1f40b832d3001e8e053249e4ed","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-8a65e165240c40e0","role":"sequence","name":"Triangular array of generalized Narayana numbers T(n,k) = 2*binomial(n+1,k)* binomial(n-2,k-1)/(n+1) for n >= 1 and 0 <= k <= n-1, read by rows.","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: 1;\n  1: 2,   2;\n  2: 2,   8,    3;\n  3: 2,  18,   20,     4;\n  4: 2,  32,   75,    40,     5;\n  5: 2,  50,  204,   220,    70,     6;\n  6: 2,  72,  455,   840,   525,   112,     7;\n  7: 2,  98,  888,  2520,  2660,  1092,   168,    8;\n  8: 2, 128, 1575,  6384, 10290,  7056,  2058,  240,   9;\n  9: 2, 162, 2600, 14280, 32760, 34104, 16464, 3600, 330, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^4+2*n^3-n^2-2*n+(2*n^3+2*n^2-4*n)*x+(n^2+2*n)*x^2)*P(n,x) = (n^4+2*n^3-n^2-2*n+(3*n^4+6*n^3-8*n^2-2*n+1)*x+(6*n^3+2*n^2-2)*x^2+(3*n^2+2*n+1)*x^3)*P(n-1,x) + ((2*n^3+4*n^2-2*n-1)*x-(2*n^3-4*n-1)*x^2-(4*n^2-1)*x^3-(2*n+1)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^4 + 2*n^3 - n^2 - 2*n + (3*n^4 + 6*n^3 - 8*n^2 - 2*n + 1)*x + (6*n^3 + 2*n^2 - 2)*x^2 + (3*n^2 + 2*n + 1)*x^3)*P[n - 1, x] + ((2*n^3 + 4*n^2 - 2*n - 1)*x - (2*n^3 - 4*n - 1)*x^2 - (4*n^2 - 1)*x^3 - (2*n + 1)*x^4)*D[P[n - 1, x], x])/(n^4 + 2*n^3 - n^2 - 2*n + (2*n^3 + 2*n^2 - 4*n)*x + (n^2 + 2*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^4 + 2*n^3 - n^2 - 2*n + (3*n^4 + 6*n^3 - 8*n^2 - 2*n + 1)*x + (6*n^3 + 2*n^2 - 2)*x^2 + (3*n^2 + 2*n + 1)*x^3)*v[i - 1] + ((2*n^3 + 4*n^2 - 2*n - 1)*x - (2*n^3 - 4*n - 1)*x^2 - (4*n^2 - 1)*x^3 - (2*n + 1)*x^4)*deriv(v[i - 1], x))/(n^4 + 2*n^3 - n^2 - 2*n + (2*n^3 + 2*n^2 - 4*n)*x + (n^2 + 2*n)*x^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-8a65e165240c40e0","steps":["D+1"],"edges":["e-3495b1ceb6c69ff8"],"text":"A281260 \u2192 D+1"},"headline":"A281260 \u2192 D+1","chain":null}