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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":"46c7c81fb68b7098c0504ea6aa4f9312e9a18273927784ec2286acd58a8937f6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-8348c590b954589f","role":"sequence","name":"one_plus_derivative of Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).","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:   9,    4;\n  2:  35,   44,     9;\n  3:  79,  216,   123,     16;\n  4: 141,  616,   738,    264,     25;\n  5: 221, 1340,  2550,   1880,    485,     36;\n  6: 319, 2484,  6555,   7640,   4005,    804,    49;\n  7: 435, 4144, 14049,  22680,  18725,   7560,  1239,    64;\n  8: 569, 6416, 26628,  55216,  63070,  39984, 13076,  1808,   81;\n  9: 721, 9396, 46188, 117264, 171990, 150696, 77196, 21168, 2529, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+5*x+10*x^2+10*x^3+5*x^4+x^5)*P(n-5,x) - (5+20*x+30*x^2+20*x^3+5*x^4)*P(n-4,x) + (10+30*x+30*x^2+10*x^3)*P(n-3,x) - (10+20*x+10*x^2)*P(n-2,x) + (5+5*x)*P(n-1,x) for n >= 5, with P(0,x) = 1, P(1,x) = 9 + 4*x, P(2,x) = 35 + 44*x + 9*x^2, P(3,x) = 79 + 216*x + 123*x^2 + 16*x^3, P(4,x) = 141 + 616*x + 738*x^2 + 264*x^3 + 25*x^4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 9 + 4*x;\nP[2, x_] := 35 + 44*x + 9*x^2;\nP[3, x_] := 79 + 216*x + 123*x^2 + 16*x^3;\nP[4, x_] := 141 + 616*x + 738*x^2 + 264*x^3 + 25*x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*P[n - 5, x] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*P[n - 4, x] + (10 + 30*x + 30*x^2 + 10*x^3)*P[n - 3, x] - (10 + 20*x + 10*x^2)*P[n - 2, x] + (5 + 5*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 4*x; v[3] = 35 + 44*x + 9*x^2; v[4] = 79 + 216*x + 123*x^2 + 16*x^3; v[5] = 141 + 616*x + 738*x^2 + 264*x^3 + 25*x^4; for(n = 5, N, my(i = n + 1); v[i] = (1 + 5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5)*v[i - 5] - (5 + 20*x + 30*x^2 + 20*x^3 + 5*x^4)*v[i - 4] + (10 + 30*x + 30*x^2 + 10*x^3)*v[i - 3] - (10 + 20*x + 10*x^2)*v[i - 2] + (5 + 5*x)*v[i - 1]); 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-e7d0ffa461d9dbc4","steps":["D+1","D+1"],"edges":["e-720363525db6d149","e-c40a306b8ce3abe8"],"text":"A125092 \u2192 D+1 \u2192 D+1"},"headline":"A125092 \u2192 D+1 \u2192 D+1","chain":null}