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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":"72a7bd340e16843411f7272154a93396f093e1b7eae42e882cdad60ebdac8d30","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-820fbf6ab98121e9","role":"sequence","name":"basis_map_A013609 of Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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:      0;\n  1:      2;\n  2:     16,       8;\n  3:     84,      96,      24;\n  4:    380,     696,     384,      64;\n  5:   1594,    4040,    3552,    1280,     160;\n  6:   6384,   20712,   25248,   14400,    3840,     384;\n  7:  24782,   98088,  153552,  121280,   50880,   10752,    896;\n  8:  94060,  439640,  842304,  855040,  495680,  163968,  28672,  2048;\n  9: 350988, 1892376, 4293648, 5342016, 3978720, 1814400, 494592, 73728, 4608;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^3-56*n^2+114*n-64-(12*n^3-8*n^2-60*n+56)*x+(16*n^2-16)*x^2)*P(n,x) = -(48*n^3-304*n^2+420*n-104+(48*n^3-512*n^2+888*n-160)*x-(160*n^2-528*n+32)*x^2+96*n*x^3)*P(n-1,x) + (39*n^2-91*n+26+(138*n^2-374*n+92)*x+(132*n^2-492*n+88)*x^2+(24*n^2-232*n+16)*x^3-32*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(48*n^3 - 304*n^2 + 420*n - 104 + (48*n^3 - 512*n^2 + 888*n - 160)*x - (160*n^2 - 528*n + 32)*x^2 + 96*n*x^3)*P[n - 1, x] + (39*n^2 - 91*n + 26 + (138*n^2 - 374*n + 92)*x + (132*n^2 - 492*n + 88)*x^2 + (24*n^2 - 232*n + 16)*x^3 - 32*n*x^4)*D[P[n - 1, x], x])/(-6*n^3 + 56*n^2 - 114*n + 64 + (12*n^3 - 8*n^2 - 60*n + 56)*x - (16*n^2 - 16)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(48*n^3 - 304*n^2 + 420*n - 104 + (48*n^3 - 512*n^2 + 888*n - 160)*x - (160*n^2 - 528*n + 32)*x^2 + 96*n*x^3)*v[i - 1] + (39*n^2 - 91*n + 26 + (138*n^2 - 374*n + 92)*x + (132*n^2 - 492*n + 88)*x^2 + (24*n^2 - 232*n + 16)*x^3 - 32*n*x^4)*deriv(v[i - 1], x))/(-6*n^3 + 56*n^2 - 114*n + 64 + (12*n^3 - 8*n^2 - 60*n + 56)*x - (16*n^2 - 16)*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-d6b7687ba0ff59dd","steps":["basis A013609","D"],"edges":["e-0d5d22b8fd97251b","e-7e683bb415cc9ba4"],"text":"A116412 \u2192 basis A013609 \u2192 D"},"headline":"A116412 \u2192 basis A013609 \u2192 D","chain":null}