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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":"bdcf07b481529df5b3c910fb65870abb5c2e158b35dc989a333ab1fe5f120d1c","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-16c98a2b049669c5","role":"sequence","name":"basis_map_A013610 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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:           19,            3;\n  2:          352,          114,            9;\n  3:         6400,         3168,          513,          27;\n  4:       114688,        76800,        19008,        2052,          81;\n  5:      2031616,      1720320,       576000,       95040,        7695,        243;\n  6:     35651584,     36569088,     15482880,     3456000,      427680,      27702,       729;\n  7:    620756992,    748683264,    383975424,   108380160,    18144000,    1796256,     96957,     2187;\n  8:  10737418240,  14898167808,   8984199168,  3071803392,   650280960,   87091200,   7185024,   332424,    6561;\n  9: 184683593728, 289910292480, 201125265408, 80857792512, 20734672896, 3511517184, 391910400, 27713664, 1121931, 19683;\n","recurrence_sentence":"The row polynomials satisfy (3*n+13+3*x)*P(n,x) = (48*n+256+(9*n+96)*x+9*x^2)*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[((48*n + 256 + (9*n + 96)*x + 9*x^2)*P[n - 1, x])/(3*n + 13 + 3*x)]];\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] = ((48*n + 256 + (9*n + 96)*x + 9*x^2)*v[i - 1])/(3*n + 13 + 3*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-ff0214c333bfdcca","steps":["basis A013610","D+1"],"edges":["e-d2b5f968b40c2621","e-c4dbc621da1d9079"],"text":"A027467 \u2192 basis A013610 \u2192 D+1"},"headline":"A027467 \u2192 basis A013610 \u2192 D+1","chain":null}