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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":"89b13f84fb6ac343d62c4957a5b4ed3bcc4e5f64b55778629129e2d1b2a7daf2","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-5426b03c17935143","role":"sequence","name":"reversal of Triangle T(n,k), 0 <= k <= n, read by rows, given by [1,0,-1,0,0,0,0,0,0,...] DELTA [3,-2,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:    4,    1;\n  2:    7,    6,     1;\n  3:   24,   29,    10,     1;\n  4:   33,   68,    46,    12,     1;\n  5:  108,  261,   228,    90,    16,    1;\n  6:  135,  450,   579,   364,   117,   18,    1;\n  7:  432, 1593,  2358,  1807,   772,  183,   22,   1;\n  8:  513, 2376,  4572,  4744,  2886, 1048,  220,  24,  1;\n  9: 1620, 8073, 17064, 20004, 14288, 6430, 1816, 308, 28, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2-n-9+(4*n^2+9*n-15)*x+(n^2+9*n-1)*x^2+(2*n+4)*x^3+x^4)*P(n,x) = -(16*n^3-40*n^2+3*n+45+(24*n^3-10*n^2-96*n+96)*x+(12*n^3+35*n^2-90*n+9)*x^2+(2*n^3+23*n^2-13*n-45)*x^3+(4*n^2+8*n-22)*x^4+(2*n-3)*x^5)*P(n-1,x) + (24*n^2+12*n-9+(56*n^2+94*n-6)*x+(46*n^2+168*n+62)*x^2+(16*n^2+118*n+114)*x^3+(2*n^2+36*n+73)*x^4+(4*n+20)*x^5+2*x^6)*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[(-(16*n^3 - 40*n^2 + 3*n + 45 + (24*n^3 - 10*n^2 - 96*n + 96)*x + (12*n^3 + 35*n^2 - 90*n + 9)*x^2 + (2*n^3 + 23*n^2 - 13*n - 45)*x^3 + (4*n^2 + 8*n - 22)*x^4 + (2*n - 3)*x^5)*P[n - 1, x] + (24*n^2 + 12*n - 9 + (56*n^2 + 94*n - 6)*x + (46*n^2 + 168*n + 62)*x^2 + (16*n^2 + 118*n + 114)*x^3 + (2*n^2 + 36*n + 73)*x^4 + (4*n + 20)*x^5 + 2*x^6)*D[P[n - 1, x], x])/(4*n^2 - n - 9 + (4*n^2 + 9*n - 15)*x + (n^2 + 9*n - 1)*x^2 + (2*n + 4)*x^3 + x^4)]];\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] = (-(16*n^3 - 40*n^2 + 3*n + 45 + (24*n^3 - 10*n^2 - 96*n + 96)*x + (12*n^3 + 35*n^2 - 90*n + 9)*x^2 + (2*n^3 + 23*n^2 - 13*n - 45)*x^3 + (4*n^2 + 8*n - 22)*x^4 + (2*n - 3)*x^5)*v[i - 1] + (24*n^2 + 12*n - 9 + (56*n^2 + 94*n - 6)*x + (46*n^2 + 168*n + 62)*x^2 + (16*n^2 + 118*n + 114)*x^3 + (2*n^2 + 36*n + 73)*x^4 + (4*n + 20)*x^5 + 2*x^6)*deriv(v[i - 1], x))/(4*n^2 - n - 9 + (4*n^2 + 9*n - 15)*x + (n^2 + 9*n - 1)*x^2 + (2*n + 4)*x^3 + x^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-7196be931c2ed996","steps":["reverse","D+1"],"edges":["e-67f427bf8180eb6f","e-394d7c824b370f52"],"text":"A152842 \u2192 reverse \u2192 D+1"},"headline":"A152842 \u2192 reverse \u2192 D+1","chain":null}