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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":"e465add3ea123c1fcc4d96d748a08fde8528af2b0c322886fbe75bf9a08c04e4","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-a5bdea4e5eb1caa8","role":"sequence","name":"shift_x_plus_one of Triangle read by rows, A007318 rows repeated three times.","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:  1;\n  3:  3,  1;\n  4:  3,  1;\n  5:  3,  1;\n  6:  8,  6, 1;\n  7:  8,  6, 1;\n  8:  8,  6, 1;\n  9: 20, 24, 9, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3+6*n^2-43*n-48+(9*n^2+9*n-156)*x+(27*n-27)*x^2+27*x^3)*P(n,x) = (2*n^3+30*n^2-176*n-84+(n^3+33*n^2-16*n-516)*x+(9*n^2+90*n-237)*x^2+(27*n+54)*x^3+27*x^4)*P(n-3,x) - (108*n-540+(108*n-432)*x+(27*n-27)*x^2+27*x^3)*d/dx P(n-3,x) + (216+324*x+162*x^2+27*x^3)*d^2/dx^2 P(n-3,x) for n >= 3, with P(0,x) = 1, P(1,x) = 1, P(2,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 30*n^2 - 176*n - 84 + (n^3 + 33*n^2 - 16*n - 516)*x + (9*n^2 + 90*n - 237)*x^2 + (27*n + 54)*x^3 + 27*x^4)*P[n - 3, x] - (108*n - 540 + (108*n - 432)*x + (27*n - 27)*x^2 + 27*x^3)*D[P[n - 3, x], x] + (216 + 324*x + 162*x^2 + 27*x^3)*D[P[n - 3, x], {x, 2}])/(n^3 + 6*n^2 - 43*n - 48 + (9*n^2 + 9*n - 156)*x + (27*n - 27)*x^2 + 27*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^3 + 30*n^2 - 176*n - 84 + (n^3 + 33*n^2 - 16*n - 516)*x + (9*n^2 + 90*n - 237)*x^2 + (27*n + 54)*x^3 + 27*x^4)*v[i - 3] - (108*n - 540 + (108*n - 432)*x + (27*n - 27)*x^2 + 27*x^3)*deriv(v[i - 3], x) + (216 + 324*x + 162*x^2 + 27*x^3)*deriv(deriv(v[i - 3], x), x))/(n^3 + 6*n^2 - 43*n - 48 + (9*n^2 + 9*n - 156)*x + (27*n - 27)*x^2 + 27*x^3)); 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-99e5bf113a0a46cd","steps":["basis A007318","D+1"],"edges":["e-131d94ab8eb56aaf","e-7949058fe27c71c2"],"text":"A152828 \u2192 basis A007318 \u2192 D+1"},"headline":"A152828 \u2192 basis A007318 \u2192 D+1","chain":null}