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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":"7bd1d8b42828c308ed03a3687d7a6911e0efb9ccc868c0eeaa065716eb19ddb3","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-45b8d33479b2056d","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","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:     6,      2;\n   3:    25,     22,       4;\n   4:    91,    142,      64,       8;\n   5:   306,    710,     552,     168,      16;\n   6:   979,   3060,    3492,    1808,     416,      32;\n   7:  3029,  11988,   18324,   13808,    5360,     992,     64;\n   8:  9150,  43978,   84936,   85320,   47920,   14880,   2304,    128;\n   9: 27149, 153818,  360632,  456584,  340560,  152352,  39424,   5248,   256;\n  10: 79439, 518930, 1434836, 2202704, 2065968, 1223712, 454720, 100864, 11776, 512;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-72*n^2+148*n-114-(16*n-24)*x-(16*n-24)*x^2)*P(n,x) = (24*n^3-40*n^2+62*n-24+(16*n^3-32*n^2-12*n+20)*x-(56*n-48)*x^2-(16*n-16)*x^3)*P(n-1,x) + (20*n^2-40*n+15+(48*n^2-96*n-24)*x+(16*n^2-32*n-152)*x^2-96*x^3-16*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[((24*n^3 - 40*n^2 + 62*n - 24 + (16*n^3 - 32*n^2 - 12*n + 20)*x - (56*n - 48)*x^2 - (16*n - 16)*x^3)*P[n - 1, x] + (20*n^2 - 40*n + 15 + (48*n^2 - 96*n - 24)*x + (16*n^2 - 32*n - 152)*x^2 - 96*x^3 - 16*x^4)*D[P[n - 1, x], x])/(16*n^3 - 72*n^2 + 148*n - 114 - (16*n - 24)*x - (16*n - 24)*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] = ((24*n^3 - 40*n^2 + 62*n - 24 + (16*n^3 - 32*n^2 - 12*n + 20)*x - (56*n - 48)*x^2 - (16*n - 16)*x^3)*v[i - 1] + (20*n^2 - 40*n + 15 + (48*n^2 - 96*n - 24)*x + (16*n^2 - 32*n - 152)*x^2 - 96*x^3 - 16*x^4)*deriv(v[i - 1], x))/(16*n^3 - 72*n^2 + 148*n - 114 - (16*n - 24)*x - (16*n - 24)*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-318700a428d7b97b","steps":["basis A013609","D+1"],"edges":["e-0d48af5cb3d93d03","e-d8a1926efb3a3bc3"],"text":"A111125 \u2192 basis A013609 \u2192 D+1"},"headline":"A111125 \u2192 basis A013609 \u2192 D+1","chain":null}