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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":"6d884bda66df799a1b2c6de7e1525ac935deda51218b6172c5b8cd4903444dc4","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-f39c1a4c1fe3654f","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","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:      0;\n   2:      1;\n   3:      5,       2;\n   4:     20,      18,       3;\n   5:     82,     118,      42,       4;\n   6:    379,     740,     405,      80,      5;\n   7:   2043,    4898,    3555,    1060,    135,     6;\n   8:  12858,   35560,   31437,   12320,   2345,   210,    7;\n   9:  93308,  286396,  292488,  139356,  34720,  4620,  308,   8;\n  10: 769053, 2556936, 2915070, 1608264, 487095, 84672, 8358, 432, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^3-15*n^2+74*n-120)*P(n-6,x) + (3*n^3-39*n^2+168*n-240+(3*n^2-27*n+60)*x)*P(n-5,x) - (3*n^3-30*n^2+97*n-100+(6*n^2-45*n+84)*x+(3*n-12)*x^2)*P(n-4,x) + (n^3-3*n^2-12*n+37+(3*n^2-12*n+6)*x+(3*n-9)*x^2+x^3)*P(n-3,x) - (3*n^2-12*n+10+(6*n-15)*x+3*x^2)*P(n-2,x) + (3*n-6+3*x)*P(n-1,x) for n >= 7, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 5 + 2*x, P(4,x) = 20 + 18*x + 3*x^2, P(5,x) = 82 + 118*x + 42*x^2 + 4*x^3, P(6,x) = 379 + 740*x + 405*x^2 + 80*x^3 + 5*x^4.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[3, x_] := 5 + 2*x;\nP[4, x_] := 20 + 18*x + 3*x^2;\nP[5, x_] := 82 + 118*x + 42*x^2 + 4*x^3;\nP[6, x_] := 379 + 740*x + 405*x^2 + 80*x^3 + 5*x^4;\nP[n_, x_] := P[n, x] = Expand[-(n^3 - 15*n^2 + 74*n - 120)*P[n - 6, x] + (3*n^3 - 39*n^2 + 168*n - 240 + (3*n^2 - 27*n + 60)*x)*P[n - 5, x] - (3*n^3 - 30*n^2 + 97*n - 100 + (6*n^2 - 45*n + 84)*x + (3*n - 12)*x^2)*P[n - 4, x] + (n^3 - 3*n^2 - 12*n + 37 + (3*n^2 - 12*n + 6)*x + (3*n - 9)*x^2 + x^3)*P[n - 3, x] - (3*n^2 - 12*n + 10 + (6*n - 15)*x + 3*x^2)*P[n - 2, x] + (3*n - 6 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 5 + 2*x; v[4] = 20 + 18*x + 3*x^2; v[5] = 82 + 118*x + 42*x^2 + 4*x^3; v[6] = 379 + 740*x + 405*x^2 + 80*x^3 + 5*x^4; for(n = 7, N, my(i = n); v[i] = -(n^3 - 15*n^2 + 74*n - 120)*v[i - 6] + (3*n^3 - 39*n^2 + 168*n - 240 + (3*n^2 - 27*n + 60)*x)*v[i - 5] - (3*n^3 - 30*n^2 + 97*n - 100 + (6*n^2 - 45*n + 84)*x + (3*n - 12)*x^2)*v[i - 4] + (n^3 - 3*n^2 - 12*n + 37 + (3*n^2 - 12*n + 6)*x + (3*n - 9)*x^2 + x^3)*v[i - 3] - (3*n^2 - 12*n + 10 + (6*n - 15)*x + 3*x^2)*v[i - 2] + (3*n - 6 + 3*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3272d5bc697b9cb4","steps":["D","D+1"],"edges":["e-d161d3edb4630d0d","e-159b2c638f83b101"],"text":"A094816 \u2192 D \u2192 D+1"},"headline":"A094816 \u2192 D \u2192 D+1","chain":null}