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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":"b096ff00bce72089e2ce37b319cfa5ac79e0eaf90e69e1538f993299deffc988","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-54284d24a094e2d3","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, 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:          0;\n  1:          1;\n  2:          1,          2;\n  3:          4,         10,          3;\n  4:         36,         98,         42,         4;\n  5:        576,       1640,        819,       120,        5;\n  6:      14400,      42152,      22935,      4092,      275,       6;\n  7:     518400,    1546272,     888888,    177892,    15015,     546,      7;\n  8:   25401600,   76804128,   45874920,   9901892,   958100,   44772,    980,    8;\n  9: 1625702400, 4966267392, 3051201072, 694887648, 73695765, 4015128, 114954, 1632, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(2*n^2-8*n+8)*x+x^2)*P(n-2,x) + (2*n^2-6*n+5+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (2*n^2 - 8*n + 8)*x + x^2)*P[n - 2, x] + (2*n^2 - 6*n + 5 + 2*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (2*n^2 - 8*n + 8)*x + x^2)*v[i - 2] + (2*n^2 - 6*n + 5 + 2*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-54284d24a094e2d3","steps":["D"],"edges":["e-b8537f99f5c9c158"],"text":"A269944 \u2192 D"},"headline":"A269944 \u2192 D","chain":null}