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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":"1726f4200279bc9a5311d225760603ad0fa3cd37b47019815363d98956dfc967","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-a06f8ee83f308dcc","role":"sequence","name":"basis_map_A013610 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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,            3;\n  2:           68,           54,            9;\n  3:          912,          960,          297,           27;\n  4:        15744,        19680,         8280,         1404,          81;\n  5:       333312,       469632,       239760,        56160,        6075,        243;\n  6:      8368128,     12902400,      7551936,      2177280,      328860,      24786,       729;\n  7:    243154944,    402628608,    261386496,     87429888,    16420320,    1741824,     96957,     2187;\n  8:   8028684288,  14095319040,   9935327232,   3722277888,   818221824,  108755136,   8573040,   367416,    6561;\n  9: 296959868928, 547676356608, 413021306880, 169388789760, 42019326720, 6581481984, 654000480, 39890880, 1358127, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(24*n-24)*x+9*x^2)*P(n-2,x) + (8*n-4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 7 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (24*n - 24)*x + 9*x^2)*P[n - 2, x] + (8*n - 4 + 6*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] = 7 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (24*n - 24)*x + 9*x^2)*v[i - 2] + (8*n - 4 + 6*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-9c5c208df8605708","steps":["basis A013610","D+1"],"edges":["e-bf3b7a758db4c427","e-ee207a1475354d5c"],"text":"A225471 \u2192 basis A013610 \u2192 D+1"},"headline":"A225471 \u2192 basis A013610 \u2192 D+1","chain":null}