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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":"0bbcbfccd2ef65b972dac8240c07297f25d68eea58e16870b09726483ecad0aa","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-bf8bba6616804a77","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:       2,       1;\n  2:       5,       3;\n  3:      16,      12,      1;\n  4:      66,      56,      7;\n  5:     336,     306,     48,     1;\n  6:    2040,    1944,    351,    13;\n  7:   14400,   14160,   2808,   140,   1;\n  8:  115920,  116640,  24720,  1484,  21;\n  9: 1048320, 1073520, 239040, 16300, 330, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - (2*n-4)*x*P(n-3,x) - (n^2-3*n+2-2*x)*P(n-2,x) + (2*n-2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 5 + 3*x, P(3,x) = 16 + 12*x + x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 5 + 3*x;\nP[3, x_] := 16 + 12*x + x^2;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - (2*n - 4)*x*P[n - 3, x] - (n^2 - 3*n + 2 - 2*x)*P[n - 2, x] + (2*n - 2)*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] = 2 + x; v[3] = 5 + 3*x; v[4] = 16 + 12*x + x^2; for(n = 4, N, my(i = n + 1); v[i] = -x^2*v[i - 4] - (2*n - 4)*x*v[i - 3] - (n^2 - 3*n + 2 - 2*x)*v[i - 2] + (2*n - 2)*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-bf8bba6616804a77","steps":["D+1"],"edges":["e-e5a7cbe2c0005a8b"],"text":"A249247 \u2192 D+1"},"headline":"A249247 \u2192 D+1","chain":null}