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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":"cf821fc0c5e58316c6c778b90e19daa7e29fbe4f3fffaffd7e72980101f8ec54","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-7e652d8a57b6dcff","role":"sequence","name":"Triangle of 4-Eulerian numbers.","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:    0;\n  1:    4;\n  2:   13,     32;\n  3:   32,    226,      192;\n  4:   71,   1062,     2463,      1024;\n  5:  150,   4180,    19410,     21540,      5120;\n  6:  309,  14940,   121530,    262980,    165345,     24576;\n  7:  628,  50382,   665400,   2449180,   2964420,   1164102,    114688;\n  8: 1267, 163706,  3341757,  19343500,  39216845,  29492538,   7708099,   524288;\n  9: 2546, 518992, 15822618, 136818584, 428736070, 535869264, 268112558, 48764872, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (n+2)*P(n,x) = -(2*n+6+(2*n^2+8*n+6)*x+(n^3+6*n^2+11*n+6)*x^2)*P(n-2,x) - ((n+3)*x+(n^2+4*n+3)*x^2-(n^2+5*n+6)*x^3)*d/dx P(n-2,x) + (3*n+7+(2*n^2+8*n+8)*x)*P(n-1,x) + ((n+2)*x-(n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 6 + (2*n^2 + 8*n + 6)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*P[n - 2, x] - ((n + 3)*x + (n^2 + 4*n + 3)*x^2 - (n^2 + 5*n + 6)*x^3)*D[P[n - 2, x], x] + (3*n + 7 + (2*n^2 + 8*n + 8)*x)*P[n - 1, x] + ((n + 2)*x - (n + 2)*x^2)*D[P[n - 1, x], x])/(n + 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 6 + (2*n^2 + 8*n + 6)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*v[i - 2] - ((n + 3)*x + (n^2 + 4*n + 3)*x^2 - (n^2 + 5*n + 6)*x^3)*deriv(v[i - 2], x) + (3*n + 7 + (2*n^2 + 8*n + 8)*x)*v[i - 1] + ((n + 2)*x - (n + 2)*x^2)*deriv(v[i - 1], x))/(n + 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-7e652d8a57b6dcff","steps":["D"],"edges":["e-da286ff8762bab32"],"text":"A144698 \u2192 D"},"headline":"A144698 \u2192 D","chain":null}