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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":"be0fa1fdd3439d86c3f1b98c7c7a57d9091794ab6ab9742d0fbbd3a2a84c0017","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b2e2c5159c22bcc9","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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:      8;\n  2:     36,      64;\n  3:    132,     480,      384;\n  4:    444,    2432,     4032,     2048;\n  5:   1420,   10368,    26304,    27648,    10240;\n  6:   4392,   40032,   137088,   221184,   168960,    49152;\n  7:  13264,  144864,   625152,  1367040,  1602560,   958464,   229376;\n  8:  39348,  500672,  2607552,  7214080, 11473920, 10518528,  5160960,  1048576;\n  9: 115116, 1671936, 10201536, 34182144, 68889600, 85671936, 64339968, 26738688, 4718592;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-60*n^2+60*n-8+(64*n^3-192*n^2+192*n-64)*x)*P(n,x) = (12*n^3-84*n^2-24*n+(112*n^3-256*n^2-48*n)*x+(128*n^3-128*n^2)*x^2)*P(n-1,x) + (5*n^2-20*n-15+(64*n^2-136*n-72)*x+(208*n^2-256*n-48)*x^2+(128*n^2-128*n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 8.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 8;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^3 - 84*n^2 - 24*n + (112*n^3 - 256*n^2 - 48*n)*x + (128*n^3 - 128*n^2)*x^2)*P[n - 1, x] + (5*n^2 - 20*n - 15 + (64*n^2 - 136*n - 72)*x + (208*n^2 - 256*n - 48)*x^2 + (128*n^2 - 128*n)*x^3)*D[P[n - 1, x], x])/(8*n^3 - 60*n^2 + 60*n - 8 + (64*n^3 - 192*n^2 + 192*n - 64)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 8; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^3 - 84*n^2 - 24*n + (112*n^3 - 256*n^2 - 48*n)*x + (128*n^3 - 128*n^2)*x^2)*v[i - 1] + (5*n^2 - 20*n - 15 + (64*n^2 - 136*n - 72)*x + (208*n^2 - 256*n - 48)*x^2 + (128*n^2 - 128*n)*x^3)*deriv(v[i - 1], x))/(8*n^3 - 60*n^2 + 60*n - 8 + (64*n^3 - 192*n^2 + 192*n - 64)*x)); 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-353fb5a029e4e669","steps":["basis A013611","D"],"edges":["e-a3f64a2b3fc7c9ce","e-0bb7034e7e768123"],"text":"A207607 \u2192 basis A013611 \u2192 D"},"headline":"A207607 \u2192 basis A013611 \u2192 D","chain":null}