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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":"7617e76811d9d6bc7b0abe41dacb01d4f71c1d7e5339c0c267a0edef394b47a2","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-313c8ccf28f71545","role":"sequence","name":"Triangle read by rows, Lah numbers with level 2.","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:            1,             1;\n   3:            4,             6,             1;\n   4:           36,            76,            23,            1;\n   5:          576,          1648,           704,           60,           1;\n   6:        14400,         54144,         29712,         3760,         125,          1;\n   7:       518400,       2494656,       1677504,       281792,       14100,        226,        1;\n   8:     25401600,     152918784,     122562432,     25720448,     1727892,      41804,      371,      1;\n   9:   1625702400,   12005397504,   11261507328,   2842711040,   240881472,    7841888,   105056,    568,   1;\n  10: 131681894400, 1172902809600, 1270850833152, 376086392064, 38638733120, 1589779648, 28647680, 233664, 825, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-9*n^3+30*n^2-44*n+24-(2*n^3-16*n^2+41*n-34)*x+(n^2-8*n+13)*x^2+x^3)*P(n-2,x) - ((2*n^3-13*n^2+27*n-18)*x-(4*n^2-18*n+19)*x^2+(2*n-5)*x^3)*d/dx P(n-2,x) - ((n^2-5*n+6)*x^2-(2*n-5)*x^3+x^4)*d^2/dx^2 P(n-2,x) + (2*n^2-7*n+7-(2*n-4)*x+x^2)*P(n-1,x) + (2*n-1)*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) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 9*n^3 + 30*n^2 - 44*n + 24 - (2*n^3 - 16*n^2 + 41*n - 34)*x + (n^2 - 8*n + 13)*x^2 + x^3)*P[n - 2, x] - ((2*n^3 - 13*n^2 + 27*n - 18)*x - (4*n^2 - 18*n + 19)*x^2 + (2*n - 5)*x^3)*D[P[n - 2, x], x] - ((n^2 - 5*n + 6)*x^2 - (2*n - 5)*x^3 + x^4)*D[P[n - 2, x], {x, 2}] + (2*n^2 - 7*n + 7 - (2*n - 4)*x + x^2)*P[n - 1, x] + (2*n - 1)*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] = 1 + x; for(n = 3, N, my(i = n); v[i] = -(n^4 - 9*n^3 + 30*n^2 - 44*n + 24 - (2*n^3 - 16*n^2 + 41*n - 34)*x + (n^2 - 8*n + 13)*x^2 + x^3)*v[i - 2] - ((2*n^3 - 13*n^2 + 27*n - 18)*x - (4*n^2 - 18*n + 19)*x^2 + (2*n - 5)*x^3)*deriv(v[i - 2], x) - ((n^2 - 5*n + 6)*x^2 - (2*n - 5)*x^3 + x^4)*deriv(deriv(v[i - 2], x), x) + (2*n^2 - 7*n + 7 - (2*n - 4)*x + x^2)*v[i - 1] + (2*n - 1)*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-313c8ccf28f71545","steps":["D+1"],"edges":["e-1a4462719f422dc4"],"text":"A390433 \u2192 D+1"},"headline":"A390433 \u2192 D+1","chain":null}