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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":"83c7071b30a785ca8eaf8332b509159f047de824c7dc20f12c7077a1c4534a39","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-d6b7687ba0ff59dd","role":"sequence","name":"Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).","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:    1;\n  2:    6,     2;\n  3:   21,    18,     3;\n  4:   60,    90,    36,     4;\n  5:  156,   342,   234,    60,     5;\n  6:  384,  1116,  1116,   480,    90,     6;\n  7:  912,  3312,  4419,  2760,   855,   126,    7;\n  8: 2112,  9216, 15480, 12900,  5760,  1386,  168,   8;\n  9: 4800, 24480, 49752, 52332, 31095, 10710, 2100, 216, 9;\n","recurrence_sentence":"The row polynomials satisfy -(6*n^3-28*n^2+42*n-20-(3*n^3+2*n^2-15*n+10)*x+(2*n^2-2)*x^2)*P(n,x) = -(12*n^3-44*n^2+48*n-16+(12*n^3-88*n^2+108*n-32)*x-(20*n^2-48*n+4)*x^2+6*n*x^3)*P(n-1,x) + ((12*n^2-20*n+8)*x+(24*n^2-48*n+16)*x^2+(3*n^2-21*n+2)*x^3-2*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n^3 - 44*n^2 + 48*n - 16 + (12*n^3 - 88*n^2 + 108*n - 32)*x - (20*n^2 - 48*n + 4)*x^2 + 6*n*x^3)*P[n - 1, x] + ((12*n^2 - 20*n + 8)*x + (24*n^2 - 48*n + 16)*x^2 + (3*n^2 - 21*n + 2)*x^3 - 2*n*x^4)*D[P[n - 1, x], x])/(-6*n^3 + 28*n^2 - 42*n + 20 + (3*n^3 + 2*n^2 - 15*n + 10)*x - (2*n^2 - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n^3 - 44*n^2 + 48*n - 16 + (12*n^3 - 88*n^2 + 108*n - 32)*x - (20*n^2 - 48*n + 4)*x^2 + 6*n*x^3)*v[i - 1] + ((12*n^2 - 20*n + 8)*x + (24*n^2 - 48*n + 16)*x^2 + (3*n^2 - 21*n + 2)*x^3 - 2*n*x^4)*deriv(v[i - 1], x))/(-6*n^3 + 28*n^2 - 42*n + 20 + (3*n^3 + 2*n^2 - 15*n + 10)*x - (2*n^2 - 2)*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-d6b7687ba0ff59dd","steps":["D"],"edges":["e-29ee48d67514cd97"],"text":"A116412 \u2192 D"},"headline":"A116412 \u2192 D","chain":null}