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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":"37bd2c8393141a4b971c0fd7788b429b7b8a3a9aa11c981202ec64c2259ce21d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-08fe62d9c34c34e9","role":"sequence","name":"basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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:      10,        6;\n  2:      64,      102,        27;\n  3:     352,     1008,       684,       108;\n  4:    1792,     7584,      9504,      3780,       405;\n  5:    8704,    49152,     93312,     69552,     18630,      1458;\n  6:   40960,   290304,    748800,    860544,    435456,     85050,      5103;\n  7:  188416,  1609728,   5280768,   8363520,   6609600,   2449440,    367416,    17496;\n  8:  851968,  8527872,  34062336,  69368832,  76412160,  44649792,  12737088,  1522152,   59049;\n  9: 3801088, 43646976, 205848576, 514805760, 735920640, 604910592, 274057344, 62355744, 6101730, 196830;\n","recurrence_sentence":"The row polynomials satisfy (6*n^3-5*n^2+n+(12*n^2-13*n)*x+6*n*x^2)*P(n,x) = (24*n^3+4*n^2-8*n+(54*n^3+21*n^2-73*n)*x+(108*n^2-75*n+21)*x^2+(54*n-18)*x^3)*P(n-1,x) - ((12*n^2+8*n)*x+(36*n^2+48*n-2)*x^2+(72*n+6)*x^3+36*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((24*n^3 + 4*n^2 - 8*n + (54*n^3 + 21*n^2 - 73*n)*x + (108*n^2 - 75*n + 21)*x^2 + (54*n - 18)*x^3)*P[n - 1, x] - ((12*n^2 + 8*n)*x + (36*n^2 + 48*n - 2)*x^2 + (72*n + 6)*x^3 + 36*x^4)*D[P[n - 1, x], x])/(6*n^3 - 5*n^2 + n + (12*n^2 - 13*n)*x + 6*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((24*n^3 + 4*n^2 - 8*n + (54*n^3 + 21*n^2 - 73*n)*x + (108*n^2 - 75*n + 21)*x^2 + (54*n - 18)*x^3)*v[i - 1] - ((12*n^2 + 8*n)*x + (36*n^2 + 48*n - 2)*x^2 + (72*n + 6)*x^3 + 36*x^4)*deriv(v[i - 1], x))/(6*n^3 - 5*n^2 + n + (12*n^2 - 13*n)*x + 6*n*x^2)); 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-2245b35ec7cd4857","steps":["basis A013610","D+1"],"edges":["e-0cc36dd83eba3acf","e-f1c684d2b7632ae6"],"text":"A091044 \u2192 basis A013610 \u2192 D+1"},"headline":"A091044 \u2192 basis A013610 \u2192 D+1","chain":null}