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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":"22c4c3d98ba53d18dfe94ef006b021b8e3832548ce2fc60f410ae0594a7772a5","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-91f141c4f7274be3","role":"sequence","name":"basis_map_A013609 of 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:       4,        2;\n   3:      24,       24,        4;\n   4:     128,      240,       96,        8;\n   5:     640,     1792,     1408,      320,       16;\n   6:    3072,    11520,    14080,     6400,      960,       32;\n   7:   14336,    67584,   113664,    82432,    24960,     2688,      64;\n   8:   65536,   372736,   802816,   817152,   401408,    87808,    7168,    128;\n   9:  294912,  1966080,  5177344,  6848512,  4792320,  1720320,  286720,  18432,   256;\n  10: 1310720, 10027008, 31260672, 51118080, 47038464, 24330240, 6709248, 884736, 46080, 512;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-11*n^2+19*n-10+(4*n^2-18*n+20)*x+(2*n-4)*x^2)*P(n,x) = (8*n^3-36*n^2+40*n+(12*n^3-42*n^2+6*n+60)*x+(24*n^2-88*n+80)*x^2+(12*n-24)*x^3)*P(n-1,x) - ((4*n^2-8*n)*x+(8*n^2-8*n-12)*x^2+(16*n-20)*x^3+8*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 36*n^2 + 40*n + (12*n^3 - 42*n^2 + 6*n + 60)*x + (24*n^2 - 88*n + 80)*x^2 + (12*n - 24)*x^3)*P[n - 1, x] - ((4*n^2 - 8*n)*x + (8*n^2 - 8*n - 12)*x^2 + (16*n - 20)*x^3 + 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 11*n^2 + 19*n - 10 + (4*n^2 - 18*n + 20)*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] = 4 + 2*x; for(n = 3, N, my(i = n); v[i] = ((8*n^3 - 36*n^2 + 40*n + (12*n^3 - 42*n^2 + 6*n + 60)*x + (24*n^2 - 88*n + 80)*x^2 + (12*n - 24)*x^3)*v[i - 1] - ((4*n^2 - 8*n)*x + (8*n^2 - 8*n - 12)*x^2 + (16*n - 20)*x^3 + 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 11*n^2 + 19*n - 10 + (4*n^2 - 18*n + 20)*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":["basis A013609","D+1"],"edges":["e-55a1228129c4451e","e-eaa6d1c3d4fdbaf0"],"text":"A086645 \u2192 basis A013609 \u2192 D+1"},"headline":"A086645 \u2192 basis A013609 \u2192 D+1","chain":null}