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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":"4e4931ba855e1c5e110b9de3f6970c9697f804e778b567acd04bba6f90b3f6bc","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-24a064d270958fff","role":"sequence","name":"veronese_section_2_0 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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:      0;\n  2:      1,      1;\n  3:      3,      3;\n  4:     11,     13,     1;\n  5:     50,     70,    10;\n  6:    274,    444,    88,    1;\n  7:   1764,   3234,   798,   21;\n  8:  13068,  26606,  7735,  326,  1;\n  9: 109584, 244152, 80892, 4680, 36;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256-(2*n^2-16*n+32)*x+x^2)*P(n-4,x) + (4*n^3-42*n^2+148*n-175-(4*n-14)*x)*P(n-3,x) - (6*n^2-36*n+55-2*x)*P(n-2,x) + (4*n-10)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 0, P(2,x) = 1 + x, P(3,x) = 3 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 0;\nP[2, x_] := 1 + x;\nP[3, x_] := 3 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (2*n^2 - 16*n + 32)*x + x^2)*P[n - 4, x] + (4*n^3 - 42*n^2 + 148*n - 175 - (4*n - 14)*x)*P[n - 3, x] - (6*n^2 - 36*n + 55 - 2*x)*P[n - 2, x] + (4*n - 10)*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] = 0; v[3] = 1 + x; v[4] = 3 + 3*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (2*n^2 - 16*n + 32)*x + x^2)*v[i - 4] + (4*n^3 - 42*n^2 + 148*n - 175 - (4*n - 14)*x)*v[i - 3] - (6*n^2 - 36*n + 55 - 2*x)*v[i - 2] + (4*n - 10)*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-b9fbfb57d9269e2f","steps":["Veronese 2","D+1"],"edges":["e-37358459ee0a1b93","e-c0892ab426ac6a0c"],"text":"A132393 \u2192 Veronese 2 \u2192 D+1"},"headline":"A132393 \u2192 Veronese 2 \u2192 D+1","chain":null}