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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":"84342f479fb8674c00e8d51b9eca43149b524670d7118298d15354e79678d87d","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-aaf95b093ddbf844","role":"sequence","name":"gamma_polynomial of Pascal-(1,5,1) array.","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:  1;\n  2:  6,    5;\n  3: 11,   10;\n  4: 16,   65,   25;\n  5: 21,  170,   75;\n  6: 26,  325,  525,   125;\n  7: 31,  530, 1750,   500;\n  8: 36,  785, 4125,  3750,  625;\n  9: 41, 1090, 8025, 15000, 3125;\n","recurrence_sentence":"The row polynomials satisfy (5*n^3-14*n^2+9*n+(20*n^2-49*n)*x+20*n*x^2)*P(n,x) = (5*n^3-9*n^2+4*n+(50*n^3-170*n^2+91*n)*x+(200*n^2-470*n+290)*x^2+(200*n-200)*x^3)*P(n-1,x) - ((5*n^2-4*n)*x+(100*n^2-60*n-9)*x^2+(400*n-160)*x^3+400*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[((5*n^3 - 9*n^2 + 4*n + (50*n^3 - 170*n^2 + 91*n)*x + (200*n^2 - 470*n + 290)*x^2 + (200*n - 200)*x^3)*P[n - 1, x] - ((5*n^2 - 4*n)*x + (100*n^2 - 60*n - 9)*x^2 + (400*n - 160)*x^3 + 400*x^4)*D[P[n - 1, x], x])/(5*n^3 - 14*n^2 + 9*n + (20*n^2 - 49*n)*x + 20*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] = ((5*n^3 - 9*n^2 + 4*n + (50*n^3 - 170*n^2 + 91*n)*x + (200*n^2 - 470*n + 290)*x^2 + (200*n - 200)*x^3)*v[i - 1] - ((5*n^2 - 4*n)*x + (100*n^2 - 60*n - 9)*x^2 + (400*n - 160)*x^3 + 400*x^4)*deriv(v[i - 1], x))/(5*n^3 - 14*n^2 + 9*n + (20*n^2 - 49*n)*x + 20*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-6ad01dc6e705cdb6","steps":["\u03b3","D+1"],"edges":["e-90e54edd477f164c","e-c23b6e6c502bdb01"],"text":"A081580 \u2192 \u03b3 \u2192 D+1"},"headline":"A081580 \u2192 \u03b3 \u2192 D+1","chain":null}