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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":"ec5297ad141c01415b36883c330201311531dc82ccc51c8e9c86608539f939ef","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-012ae01d9db4fd27","role":"sequence","name":"theta_plus_one of Triangle read by rows: T(n,k) = binomial(n+1,k+1)*binomial(2*n-k+1,k)/(n+1), 0<=k<=n.","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:   0;\n   2:   2;\n   3:   8,    6;\n   4:  18,   60,     12;\n   5:  32,  252,    240,     20;\n   6:  50,  720,   1680,    700,      30;\n   7:  72, 1650,   7200,   7560,    1680,     42;\n   8:  98, 3276,  23100,  46200,   26460,   3528,     56;\n   9: 128, 5880,  61152, 200200,  221760,  77616,   6720,    72;\n  10: 162, 9792, 141120, 687960, 1261260, 864864, 199584, 11880, 90;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-26*n^3+62*n^2-64*n+24)*P(n,x) = (4*n^4-18*n^3+30*n^2-22*n+6+(36*n^4-272*n^3+752*n^2-900*n+396)*x)*P(n-1,x) + ((8*n^3-42*n^2+70*n-38)*x-(52*n^3-294*n^2+516*n-292)*x^2)*d/dx P(n-1,x) - ((5*n^2-14*n+10)*x^2-(20*n^2-56*n+40)*x^3)*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^4 - 18*n^3 + 30*n^2 - 22*n + 6 + (36*n^4 - 272*n^3 + 752*n^2 - 900*n + 396)*x)*P[n - 1, x] + ((8*n^3 - 42*n^2 + 70*n - 38)*x - (52*n^3 - 294*n^2 + 516*n - 292)*x^2)*D[P[n - 1, x], x] - ((5*n^2 - 14*n + 10)*x^2 - (20*n^2 - 56*n + 40)*x^3)*D[P[n - 1, x], {x, 2}])/(4*n^4 - 26*n^3 + 62*n^2 - 64*n + 24)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((4*n^4 - 18*n^3 + 30*n^2 - 22*n + 6 + (36*n^4 - 272*n^3 + 752*n^2 - 900*n + 396)*x)*v[i - 1] + ((8*n^3 - 42*n^2 + 70*n - 38)*x - (52*n^3 - 294*n^2 + 516*n - 292)*x^2)*deriv(v[i - 1], x) - ((5*n^2 - 14*n + 10)*x^2 - (20*n^2 - 56*n + 40)*x^3)*deriv(deriv(v[i - 1], x), x))/(4*n^4 - 26*n^3 + 62*n^2 - 64*n + 24)); 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-5c0c5bce3aee8449","steps":["\u03b8+1","D"],"edges":["e-79b5bc170ae8a162","e-5607ab27daf93cd1"],"text":"A391045 \u2192 \u03b8+1 \u2192 D"},"headline":"A391045 \u2192 \u03b8+1 \u2192 D","chain":null}