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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":"dba0cda991a14ab2341c327f2e3a58ee2ec66fee8cbd21209f7a6c960418d608","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-991b9bffcf8ddedb","role":"sequence","name":"Triangle T(n,k) (n >= 1, 0 <= k <= floor((n-1)/2)) read by rows, where T(n,k) = (k+1)T(n-1,k) + (2n-4k)T(n-1,k-1).","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:    1;\n  2:    3,     2;\n  3:    9,     8;\n  4:   23,    54,     16;\n  5:   53,   324,    136;\n  6:  115,  1554,   1536,    272;\n  7:  241,  6384,  14976,   3968;\n  8:  495, 23726, 114528,  66048,   7936;\n  9: 1005, 82556, 731208, 937728, 176896;\n","recurrence_sentence":"The row polynomials satisfy (2*n-5+8*x)*P(n,x) = -(4*n-6+(8*n^2-46*n+67)*x+(8*n^3-64*n^2+190*n-256)*x^2+(48*n^2-264*n+368)*x^3+(64*n-128)*x^4)*P(n-2,x) - ((2*n-3)*x+(4*n^2-24*n+35)*x^2-(16*n^2-88*n+144)*x^3-(96*n-240)*x^4-128*x^5)*d/dx P(n-2,x) + (6*n-13+(8*n^2-46*n+89)*x+(40*n-132)*x^2+32*x^3)*P(n-1,x) + ((2*n-5)*x-(8*n-28)*x^2-32*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 6 + (8*n^2 - 46*n + 67)*x + (8*n^3 - 64*n^2 + 190*n - 256)*x^2 + (48*n^2 - 264*n + 368)*x^3 + (64*n - 128)*x^4)*P[n - 2, x] - ((2*n - 3)*x + (4*n^2 - 24*n + 35)*x^2 - (16*n^2 - 88*n + 144)*x^3 - (96*n - 240)*x^4 - 128*x^5)*D[P[n - 2, x], x] + (6*n - 13 + (8*n^2 - 46*n + 89)*x + (40*n - 132)*x^2 + 32*x^3)*P[n - 1, x] + ((2*n - 5)*x - (8*n - 28)*x^2 - 32*x^3)*D[P[n - 1, x], x])/(2*n - 5 + 8*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n - 6 + (8*n^2 - 46*n + 67)*x + (8*n^3 - 64*n^2 + 190*n - 256)*x^2 + (48*n^2 - 264*n + 368)*x^3 + (64*n - 128)*x^4)*v[i - 2] - ((2*n - 3)*x + (4*n^2 - 24*n + 35)*x^2 - (16*n^2 - 88*n + 144)*x^3 - (96*n - 240)*x^4 - 128*x^5)*deriv(v[i - 2], x) + (6*n - 13 + (8*n^2 - 46*n + 89)*x + (40*n - 132)*x^2 + 32*x^3)*v[i - 1] + ((2*n - 5)*x - (8*n - 28)*x^2 - 32*x^3)*deriv(v[i - 1], x))/(2*n - 5 + 8*x)); 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-ff270bcb802a85af","steps":["\u03b3 (shift +1)","D+1"],"edges":["e-e49ea542bfed6bbf","e-2394f256e051d5b6"],"text":"A008292 \u2192 \u03b3 (shift +1) \u2192 D+1"},"headline":"A008292 \u2192 \u03b3 (shift +1) \u2192 D+1","chain":null}