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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":"3861d5c80d411f648be07e730e60251e731f9da635ed5704ba9034b897c34aa4","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-52d9fa43425cb9e1","role":"sequence","name":"basis_map_A013609 of Coefficients of raising factorial polynomials, T(n,k) = [x^k] p(x, n) where p(x, n) = (m*x + n - 1)*p(x, n - 1) with p[x, 0] = 1, p[x, -1] = 0, p[x, 1] = m*x and m = 2. Triangle read by rows, for n >= 0 and 0 <= k <= n.","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:        6,         4;\n  2:       26,        52,        16;\n  3:      128,       392,       336,        64;\n  4:      736,      2888,      3824,      1920,       256;\n  5:     4896,     22736,     39040,     30400,     10240,      1024;\n  6:    37152,    195440,    401344,    408640,    213760,     52224,      4096;\n  7:   317376,   1840576,   4319168,   5300736,   3646720,   1383424,    258048,    16384;\n  8:  3017664,  18942912,  49361856,  69699840,  58145024,  29130752,   8429568,  1245184,   65536;\n  9: 31628160, 212118912, 601966464, 949402880, 916210176, 559211520, 214351872, 49053696, 5898240, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2+8*n*x+16*x^2)*P(n-2,x) + (2*n+1+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 + 8*n*x + 16*x^2)*P[n - 2, x] + (2*n + 1 + 8*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 8*n*x + 16*x^2)*v[i - 2] + (2*n + 1 + 8*x)*v[i - 1]); 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-60c7e36cbe887b7e","steps":["basis A013609","D+1"],"edges":["e-7070cc8d62757598","e-52e4879057a1e56d"],"text":"A137320 \u2192 basis A013609 \u2192 D+1"},"headline":"A137320 \u2192 basis A013609 \u2192 D+1","chain":null}