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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":"3a9b18c2d3e2f77a287420eed2bf2494dd8c951e6a32545a7813618060b2949c","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-9f865b2dcffd9592","role":"sequence","name":"theta_plus_one of Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","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: 3, 2;\n  2: 0, 6,  3;\n  3: 0, 6, 15,  4;\n  4: 0, 0,  0, 20,  5;\n  5: 0, 0,  0, 20, 35,  6;\n  6: 0, 0,  0,  0,  0, 42,  7;\n  7: 0, 0,  0,  0,  0, 42, 63,  8;\n  8: 0, 0,  0,  0,  0,  0,  0, 72,  9;\n  9: 0, 0,  0,  0,  0,  0,  0, 72, 99, 10;\n","recurrence_sentence":"The row polynomials satisfy (n^4-4*n^3+5*n^2-2*n+(2*n^3-6*n^2+4*n)*x+(n^2-n)*x^2)*P(n,x) = -(n^5-7*n^4+17*n^3-17*n^2+6*n+(2*n^5-7*n^4+19*n^2-14*n)*x+(4*n^4-11*n^3+4*n^2+9*n-6)*x^2+(2*n^3-3*n^2-n+2)*x^3)*P(n-1,x) + ((n^4-4*n^3+5*n^2-2*n)*x+(2*n^4-2*n^3-5*n^2+5*n)*x^2+(4*n^3-3*n^2-3*n+2)*x^3+(2*n^2-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^5 - 7*n^4 + 17*n^3 - 17*n^2 + 6*n + (2*n^5 - 7*n^4 + 19*n^2 - 14*n)*x + (4*n^4 - 11*n^3 + 4*n^2 + 9*n - 6)*x^2 + (2*n^3 - 3*n^2 - n + 2)*x^3)*P[n - 1, x] + ((n^4 - 4*n^3 + 5*n^2 - 2*n)*x + (2*n^4 - 2*n^3 - 5*n^2 + 5*n)*x^2 + (4*n^3 - 3*n^2 - 3*n + 2)*x^3 + (2*n^2 - 1)*x^4)*D[P[n - 1, x], x])/(n^4 - 4*n^3 + 5*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x + (n^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; v[2] = 3 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^5 - 7*n^4 + 17*n^3 - 17*n^2 + 6*n + (2*n^5 - 7*n^4 + 19*n^2 - 14*n)*x + (4*n^4 - 11*n^3 + 4*n^2 + 9*n - 6)*x^2 + (2*n^3 - 3*n^2 - n + 2)*x^3)*v[i - 1] + ((n^4 - 4*n^3 + 5*n^2 - 2*n)*x + (2*n^4 - 2*n^3 - 5*n^2 + 5*n)*x^2 + (4*n^3 - 3*n^2 - 3*n + 2)*x^3 + (2*n^2 - 1)*x^4)*deriv(v[i - 1], x))/(n^4 - 4*n^3 + 5*n^2 - 2*n + (2*n^3 - 6*n^2 + 4*n)*x + (n^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-ea2139b8dbf6bf8d","steps":["\u03b8+1","D+1"],"edges":["e-246a0576b6d0a23b","e-f647e3f1d58a9809"],"text":"A133080 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A133080 \u2192 \u03b8+1 \u2192 D+1","chain":null}