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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":"e075ba27a742ce5ec0570e2bd1a2c88c487c4189f1c3a2a76ab83a7840a45520","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-f7935c36a3c0ec8c","role":"sequence","name":"Triangle read by rows: T(n, k) = binomial(2n, 2k).","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:   2,    1;\n   3:   7,    8,     1;\n   4:  16,   45,    18,      1;\n   5:  29,  168,   154,     32,      1;\n   6:  46,  465,   840,    390,     50,      1;\n   7:  67, 1056,  3267,   2904,    825,     72,     1;\n   8:  92, 2093, 10010,  15015,   8008,   1547,    98,    1;\n   9: 121, 3760, 25844,  59488,  52910,  18928,  2660,  128,   1;\n  10: 154, 6273, 58752, 193596, 262548, 155142, 39984, 4284, 162, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-13*n^2+29*n-22+(4*n^2-21*n+26)*x+(2*n-4)*x^2)*P(n,x) = (2*n^3-9*n^2+14*n-8+(6*n^3-31*n^2+39*n-2)*x+(12*n^2-55*n+62)*x^2+(6*n-12)*x^3)*P(n-1,x) + ((4*n^2-12*n+6)*x-(4*n^2-20*n+24)*x^2-(8*n-22)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 9*n^2 + 14*n - 8 + (6*n^3 - 31*n^2 + 39*n - 2)*x + (12*n^2 - 55*n + 62)*x^2 + (6*n - 12)*x^3)*P[n - 1, x] + ((4*n^2 - 12*n + 6)*x - (4*n^2 - 20*n + 24)*x^2 - (8*n - 22)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 13*n^2 + 29*n - 22 + (4*n^2 - 21*n + 26)*x + (2*n - 4)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; for(n = 3, N, my(i = n); v[i] = ((2*n^3 - 9*n^2 + 14*n - 8 + (6*n^3 - 31*n^2 + 39*n - 2)*x + (12*n^2 - 55*n + 62)*x^2 + (6*n - 12)*x^3)*v[i - 1] + ((4*n^2 - 12*n + 6)*x - (4*n^2 - 20*n + 24)*x^2 - (8*n - 22)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 13*n^2 + 29*n - 22 + (4*n^2 - 21*n + 26)*x + (2*n - 4)*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-f7935c36a3c0ec8c","steps":["D+1"],"edges":["e-792191c5a125b519"],"text":"A086645 \u2192 D+1"},"headline":"A086645 \u2192 D+1","chain":null}