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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":"f517af3908aa4d706649970184174f49fcaba3b4688ec655672ade6378ecde1d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9401e1a9db1eb1a5","role":"sequence","name":"Triangle of Legendre-Stirling numbers of the second kind T(n,j), n >= 1, 1 <= j <= n, read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       3,         1;\n   3:      12,        10,         1;\n   4:      60,        92,        23,         1;\n   5:     336,       904,       412,        44,         1;\n   6:    1968,      9584,      7100,      1372,        75,        1;\n   7:   11712,    107296,    124816,     38432,      3752,      118,       1;\n   8:   70080,   1241024,   2268848,   1046800,    160904,     8904,     175,     1;\n   9:  420096,  14612608,  42519232,  28622912,   6469104,   556880,   18984,   248,   1;\n  10: 2519808, 173672192, 815322560, 794938048, 254083216, 31764656, 1664168, 37224, 339, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(12-8*x-2*x^2+x^3)*P(n-2,x) - (12*x-15*x^2+4*x^3)*d/dx P(n-2,x) - (2*x^2-3*x^3+x^4)*d^2/dx^2 P(n-2,x) + (8-4*x+x^2)*P(n-1,x) + 6*x*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[-(12 - 8*x - 2*x^2 + x^3)*P[n - 2, x] - (12*x - 15*x^2 + 4*x^3)*D[P[n - 2, x], x] - (2*x^2 - 3*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (8 - 4*x + x^2)*P[n - 1, x] + 6*x*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = -(12 - 8*x - 2*x^2 + x^3)*v[i - 2] - (12*x - 15*x^2 + 4*x^3)*deriv(v[i - 2], x) - (2*x^2 - 3*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (8 - 4*x + x^2)*v[i - 1] + 6*x*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9401e1a9db1eb1a5","steps":["D+1"],"edges":["e-4ae58b8c2d15c01e"],"text":"A071951 \u2192 D+1"},"headline":"A071951 \u2192 D+1","chain":null}