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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":"6fe1c9b03625758befe9726ee2a8a1c76989a39b51a3b77b8ebf81ce567de785","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f13c6b611f4ce3bb","role":"sequence","name":"Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","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:       0;\n  1:       0;\n  2:       3;\n  3:      20;\n  4:     130,      30;\n  5:     924,     420;\n  6:    7308,    4760,     315;\n  7:   64224,   52864,    7560;\n  8:  623376,  607320,  132300,   3780;\n  9: 6636960, 7357680, 2115960, 138600;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = -(n^3-n)*x^2*P(n-4,x) - (2*n^3-2*n)*x*P(n-3,x) - (n^3-n-(2*n^2+2*n)*x)*P(n-2,x) + (2*n^2+n-1)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 3, P(3,x) = 20.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 3;\nP[3, x_] := 20;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n)*x^2*P[n - 4, x] - (2*n^3 - 2*n)*x*P[n - 3, x] - (n^3 - n - (2*n^2 + 2*n)*x)*P[n - 2, x] + (2*n^2 + n - 1)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 3; v[4] = 20; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 - n)*x^2*v[i - 4] - (2*n^3 - 2*n)*x*v[i - 3] - (n^3 - n - (2*n^2 + 2*n)*x)*v[i - 2] + (2*n^2 + n - 1)*v[i - 1])/(n)); 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-f13c6b611f4ce3bb","steps":["D"],"edges":["e-24ec188b504c13b5"],"text":"A008306 \u2192 D"},"headline":"A008306 \u2192 D","chain":null}