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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":"a50743bed3353772f98c50b1b2354b9d61b2127bff7f3e81793c4afa4fc7c5fe","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-146d64c3583c8509","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).","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:       7,        4;\n  2:      39,       60,        16;\n  3:     213,      552,       384,        64;\n  4:    1165,     4296,      5120,      2176,       256;\n  5:    6403,    30780,     52960,     39040,     11520,      1024;\n  6:   35371,   210132,    476400,    512960,    264960,     58368,      4096;\n  7:  196297,  1390480,   3927168,   5622400,   4282880,   1662976,    286720,    16384;\n  8: 1093785,  9004560,  30510592,  54795776,  55931904,  32284672,   9863168,  1376256,   65536;\n  9: 6116031, 57403764, 227148480, 492086784, 633959424, 493645824, 225705984, 56033280, 6488064, 262144;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3+16*n^2-3*n-6+(16*n^2+22*n-20)*x+(8*n+16)*x^2)*P(n,x) = (56*n^3+64*n^2-9*n-6+(96*n^3+176*n^2-66*n-20)*x+(192*n^2+32*n+16)*x^2+96*n*x^3)*P(n-1,x) - (8*n^2+12*n+3+(48*n^2+88*n+24)*x+(64*n^2+192*n+68)*x^2+(128*n+96)*x^3+64*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[((56*n^3 + 64*n^2 - 9*n - 6 + (96*n^3 + 176*n^2 - 66*n - 20)*x + (192*n^2 + 32*n + 16)*x^2 + 96*n*x^3)*P[n - 1, x] - (8*n^2 + 12*n + 3 + (48*n^2 + 88*n + 24)*x + (64*n^2 + 192*n + 68)*x^2 + (128*n + 96)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(8*n^3 + 16*n^2 - 3*n - 6 + (16*n^2 + 22*n - 20)*x + (8*n + 16)*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] = ((56*n^3 + 64*n^2 - 9*n - 6 + (96*n^3 + 176*n^2 - 66*n - 20)*x + (192*n^2 + 32*n + 16)*x^2 + 96*n*x^3)*v[i - 1] - (8*n^2 + 12*n + 3 + (48*n^2 + 88*n + 24)*x + (64*n^2 + 192*n + 68)*x^2 + (128*n + 96)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(8*n^3 + 16*n^2 - 3*n - 6 + (16*n^2 + 22*n - 20)*x + (8*n + 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-8ce61d302cec295b","steps":["basis A013609","D+1"],"edges":["e-01612a21c966f541","e-81072188a762f862"],"text":"A114687 \u2192 basis A013609 \u2192 D+1"},"headline":"A114687 \u2192 basis A013609 \u2192 D+1","chain":null}