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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":"1d45e224843ca9478394796d300112d8891103b2860920b32aa2316ec00c9943","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-6115e40a0e752de9","role":"sequence","name":"one_plus_derivative of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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:  2,   1;\n   3:  6,   6,    1;\n   4: 12,  24,   10,     1;\n   5: 20,  70,   56,    14,     1;\n   6: 28, 158,  216,   102,    18,    1;\n   7: 36, 294,  628,   494,   162,   22,    1;\n   8: 44, 478, 1472,  1770,   948,  236,   26,   1;\n   9: 52, 710, 2940,  5038,  4052, 1622,  324,  30,  1;\n  10: 60, 990, 5224, 12026, 13620, 8070, 2560, 426, 34, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x^3*P(n-6,x) + (3*x^2+3*x^3)*P(n-5,x) + (3*x+3*x^2+3*x^3)*P(n-4,x) + (1-3*x-3*x^2+x^3)*P(n-3,x) - (3+3*x+3*x^2)*P(n-2,x) + (3+3*x)*P(n-1,x) for n >= 7, with P(1,x) = 1, P(2,x) = 2 + x, P(3,x) = 6 + 6*x + x^2, P(4,x) = 12 + 24*x + 10*x^2 + x^3, P(5,x) = 20 + 70*x + 56*x^2 + 14*x^3 + x^4, P(6,x) = 28 + 158*x + 216*x^2 + 102*x^3 + 18*x^4 + x^5.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + x;\nP[3, x_] := 6 + 6*x + x^2;\nP[4, x_] := 12 + 24*x + 10*x^2 + x^3;\nP[5, x_] := 20 + 70*x + 56*x^2 + 14*x^3 + x^4;\nP[6, x_] := 28 + 158*x + 216*x^2 + 102*x^3 + 18*x^4 + x^5;\nP[n_, x_] := P[n, x] = Expand[x^3*P[n - 6, x] + (3*x^2 + 3*x^3)*P[n - 5, x] + (3*x + 3*x^2 + 3*x^3)*P[n - 4, x] + (1 - 3*x - 3*x^2 + x^3)*P[n - 3, x] - (3 + 3*x + 3*x^2)*P[n - 2, x] + (3 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + x; v[3] = 6 + 6*x + x^2; v[4] = 12 + 24*x + 10*x^2 + x^3; v[5] = 20 + 70*x + 56*x^2 + 14*x^3 + x^4; v[6] = 28 + 158*x + 216*x^2 + 102*x^3 + 18*x^4 + x^5; for(n = 7, N, my(i = n); v[i] = x^3*v[i - 6] + (3*x^2 + 3*x^3)*v[i - 5] + (3*x + 3*x^2 + 3*x^3)*v[i - 4] + (1 - 3*x - 3*x^2 + x^3)*v[i - 3] - (3 + 3*x + 3*x^2)*v[i - 2] + (3 + 3*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b0fc620949fb5990","steps":["D+1","D+1"],"edges":["e-87cb78b62855bfe8","e-aa62cde8d808c846"],"text":"A122542 \u2192 D+1 \u2192 D+1"},"headline":"A122542 \u2192 D+1 \u2192 D+1","chain":null}