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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":"faaa4a99b8fe9cb0fda4e430cbddef4cad492537a280436190a40275511af4de","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-eac80c2bf8e566f5","role":"sequence","name":"Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:  4,   3;\n  2:  7,  12,    3;\n  3: 10,  39,   24,     3;\n  4: 13,  84,  108,    36,     3;\n  5: 16, 147,  336,   222,    48,     3;\n  6: 19, 228,  789,   912,   381,    60,    3;\n  7: 22, 327, 1548,  2781,  1974,   585,   72,    3;\n  8: 25, 444, 2694,  6828,  7500,  3684,  834,   84,  3;\n  9: 28, 579, 4308, 14376, 22548, 16890, 6204, 1128, 96, 3;\n","recurrence_sentence":"The row polynomials satisfy -(3*n^3+n^2-2*n-(3*n^3-9*n^2+n+13)*x-(9*n^2-22*n-3)*x^2-(n^2+6*n-9)*x^3-(n-1)*x^4)*P(n,x) = -(3*n^3+4*n^2+n+(15*n^3-28*n-13)*x+(45*n^2-8*n+3)*x^2+(5*n^2+42*n+9)*x^3+(5*n+1)*x^4)*P(n-1,x) + ((3*n^2+4*n+1)*x+(30*n^2+50*n+16)*x^2+(3*n^2+105*n+71)*x^3+(20*n+107)*x^4+(n+20)*x^5+x^6)*d/dx 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[(-(3*n^3 + 4*n^2 + n + (15*n^3 - 28*n - 13)*x + (45*n^2 - 8*n + 3)*x^2 + (5*n^2 + 42*n + 9)*x^3 + (5*n + 1)*x^4)*P[n - 1, x] + ((3*n^2 + 4*n + 1)*x + (30*n^2 + 50*n + 16)*x^2 + (3*n^2 + 105*n + 71)*x^3 + (20*n + 107)*x^4 + (n + 20)*x^5 + x^6)*D[P[n - 1, x], x])/(-3*n^3 - n^2 + 2*n + (3*n^3 - 9*n^2 + n + 13)*x + (9*n^2 - 22*n - 3)*x^2 + (n^2 + 6*n - 9)*x^3 + (n - 1)*x^4)]];\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] = (-(3*n^3 + 4*n^2 + n + (15*n^3 - 28*n - 13)*x + (45*n^2 - 8*n + 3)*x^2 + (5*n^2 + 42*n + 9)*x^3 + (5*n + 1)*x^4)*v[i - 1] + ((3*n^2 + 4*n + 1)*x + (30*n^2 + 50*n + 16)*x^2 + (3*n^2 + 105*n + 71)*x^3 + (20*n + 107)*x^4 + (n + 20)*x^5 + x^6)*deriv(v[i - 1], x))/(-3*n^3 - n^2 + 2*n + (3*n^3 - 9*n^2 + n + 13)*x + (9*n^2 - 22*n - 3)*x^2 + (n^2 + 6*n - 9)*x^3 + (n - 1)*x^4)); 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-eac80c2bf8e566f5","steps":["D+1"],"edges":["e-747d16c15ece4cbb"],"text":"A249250 \u2192 D+1"},"headline":"A249250 \u2192 D+1","chain":null}