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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":"7dd1a9e3d0b9d8c0852dbc52a7bfec294d3f2872e1741219cd9cf93aa4a8bf2a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0444a4f352052985","role":"sequence","name":"basis_map_A007318 of T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","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:        5,        2;\n   3:       31,       30,         6;\n   4:      195,      312,       150,        20;\n   5:     1221,     2760,      2190,       700,        70;\n   6:     7593,    22290,     24990,     13160,      3150,       252;\n   7:    46915,   169806,    247380,    183960,     72450,     13860,      924;\n   8:   288263,  1242080,   2232468,   2157120,   1198890,    376992,    60060,    3432;\n   9:  1762825,  8817696,  18878580,  22508640,  16250850,   7207200,  1885884,  257400,   12870;\n  10: 10736973, 61178130, 152133300, 216114360, 192522330, 110882772, 40900860, 9163440, 1093950, 48620;\n","recurrence_sentence":"The row polynomials satisfy (n^3-4*n^2+6*n-3-(n-1)*x-(n-1)*x^2)*P(n,x) = (3*n^3-6*n^2+2*n+1+(2*n^3-4*n^2-5*n+7)*x-(7*n-7)*x^2-(2*n-2)*x^3)*P(n-1,x) + (4*n^2-8*n-4+(6*n^2-12*n-18)*x+(2*n^2-4*n-24)*x^2-12*x^3-2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((3*n^3 - 6*n^2 + 2*n + 1 + (2*n^3 - 4*n^2 - 5*n + 7)*x - (7*n - 7)*x^2 - (2*n - 2)*x^3)*P[n - 1, x] + (4*n^2 - 8*n - 4 + (6*n^2 - 12*n - 18)*x + (2*n^2 - 4*n - 24)*x^2 - 12*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 - 4*n^2 + 6*n - 3 - (n - 1)*x - (n - 1)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((3*n^3 - 6*n^2 + 2*n + 1 + (2*n^3 - 4*n^2 - 5*n + 7)*x - (7*n - 7)*x^2 - (2*n - 2)*x^3)*v[i - 1] + (4*n^2 - 8*n - 4 + (6*n^2 - 12*n - 18)*x + (2*n^2 - 4*n - 24)*x^2 - 12*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 - 4*n^2 + 6*n - 3 - (n - 1)*x - (n - 1)*x^2)); 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-4689fb1386d440ad","steps":["x \u21a6 x+1","D+1"],"edges":["e-437a2dd956a9df17","e-789f05df131a3a60"],"text":"A063007 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A063007 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}