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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":"91249094a35e7ea9695a23bb9b72c52465968f903c2bf92a51d08ee64164494d","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-f8bb6517bc8332de","role":"sequence","name":"theta_plus_one of A Jacobsthal triangle.","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: 4, 10,   3;\n  3: 0, 24,  24,   4;\n  4: 0, 24,  84,  44,    5;\n  5: 0,  0, 144, 208,   70,    6;\n  6: 0,  0,  96, 512,  420,  102,    7;\n  7: 0,  0,   0, 640, 1360,  744,  140,    8;\n  8: 0,  0,   0, 320, 2480, 3000, 1204,  184,   9;\n  9: 0,  0,   0,   0, 2400, 7200, 5824, 1824, 234, 10;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-2*n^2+(6*n^2-10*n)*x+(n^2+4*n)*x^2+n*x^3)*P(n,x) = -(8*n^3-32*n^2+24*n+(16*n^2-56*n+48)*x+(4*n^2+2*n-8)*x^2+(2*n-14)*x^3-2*x^4)*P(n-1,x) + ((16*n^2-16*n)*x+(2*n^2+46*n-32)*x^2+(14*n+44)*x^3+(n+14)*x^4+x^5)*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[(-(8*n^3 - 32*n^2 + 24*n + (16*n^2 - 56*n + 48)*x + (4*n^2 + 2*n - 8)*x^2 + (2*n - 14)*x^3 - 2*x^4)*P[n - 1, x] + ((16*n^2 - 16*n)*x + (2*n^2 + 46*n - 32)*x^2 + (14*n + 44)*x^3 + (n + 14)*x^4 + x^5)*D[P[n - 1, x], x])/(2*n^3 - 2*n^2 + (6*n^2 - 10*n)*x + (n^2 + 4*n)*x^2 + n*x^3)]];\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] = (-(8*n^3 - 32*n^2 + 24*n + (16*n^2 - 56*n + 48)*x + (4*n^2 + 2*n - 8)*x^2 + (2*n - 14)*x^3 - 2*x^4)*v[i - 1] + ((16*n^2 - 16*n)*x + (2*n^2 + 46*n - 32)*x^2 + (14*n + 44)*x^3 + (n + 14)*x^4 + x^5)*deriv(v[i - 1], x))/(2*n^3 - 2*n^2 + (6*n^2 - 10*n)*x + (n^2 + 4*n)*x^2 + n*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-ba196dcd01800741","steps":["\u03b8+1","D+1"],"edges":["e-ecb19626d7d3a316","e-b5306765e58da353"],"text":"A113953 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A113953 \u2192 \u03b8+1 \u2192 D+1","chain":null}