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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":"e9012aa8025d743b72260051b1e2b6a61287237c730804025d5fd103f969f80b","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-de2147e0b2be5082","role":"sequence","name":"basis_map_A013610 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:     1;\n  1:     9,       6;\n  2:    35,      63,      18;\n  3:   120,     369,     297,       54;\n  4:   388,    1701,    2385,     1215,      162;\n  5:  1209,    6897,   14013,    12447,     4617,      486;\n  6:  3671,   25812,   69093,    89262,    57591,    16767,     1458;\n  7: 10935,   91434,  304479,   520452,   488430,   246645,    59049,    4374;\n  8: 32095,  311139, 1240875,  2649267,  3293460,  2416635,  1000917,  203391,  13122;\n  9: 93102, 1026801, 4774005, 12250035, 19051686, 18517329, 11129643, 3903795, 688905, 39366;\n","recurrence_sentence":"The row polynomials satisfy (54*n^3-405*n^2+267*n+21+(324*n^3-972*n^2+576*n+72)*x-(378*n-459)*x^2-(324*n-324)*x^3)*P(n,x) = (81*n^3-567*n^2-414*n+333+(567*n^3-1296*n^2-1476*n+1827)*x+(486*n^3-486*n^2-2214*n+2943)*x^2-(1539*n-2025)*x^3-(486*n-486)*x^4)*P(n-1,x) + (45*n^2-180*n-340+(432*n^2-918*n-2184)*x+(1053*n^2-1296*n-4923)*x^2+(486*n^2-486*n-5076)*x^3-2511*x^4-486*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[((81*n^3 - 567*n^2 - 414*n + 333 + (567*n^3 - 1296*n^2 - 1476*n + 1827)*x + (486*n^3 - 486*n^2 - 2214*n + 2943)*x^2 - (1539*n - 2025)*x^3 - (486*n - 486)*x^4)*P[n - 1, x] + (45*n^2 - 180*n - 340 + (432*n^2 - 918*n - 2184)*x + (1053*n^2 - 1296*n - 4923)*x^2 + (486*n^2 - 486*n - 5076)*x^3 - 2511*x^4 - 486*x^5)*D[P[n - 1, x], x])/(54*n^3 - 405*n^2 + 267*n + 21 + (324*n^3 - 972*n^2 + 576*n + 72)*x - (378*n - 459)*x^2 - (324*n - 324)*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] = ((81*n^3 - 567*n^2 - 414*n + 333 + (567*n^3 - 1296*n^2 - 1476*n + 1827)*x + (486*n^3 - 486*n^2 - 2214*n + 2943)*x^2 - (1539*n - 2025)*x^3 - (486*n - 486)*x^4)*v[i - 1] + (45*n^2 - 180*n - 340 + (432*n^2 - 918*n - 2184)*x + (1053*n^2 - 1296*n - 4923)*x^2 + (486*n^2 - 486*n - 5076)*x^3 - 2511*x^4 - 486*x^5)*deriv(v[i - 1], x))/(54*n^3 - 405*n^2 + 267*n + 21 + (324*n^3 - 972*n^2 + 576*n + 72)*x - (378*n - 459)*x^2 - (324*n - 324)*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-353fb5a029e4e669","steps":["basis A013610","D+1"],"edges":["e-4d45de18adc3c05d","e-d5058458d95fe415"],"text":"A207607 \u2192 basis A013610 \u2192 D+1"},"headline":"A207607 \u2192 basis A013610 \u2192 D+1","chain":null}