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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":"fa61e322e4aa73ec590f99b9d4ab57d10eeed2d3ce6f03fde28a44f38fa322dd","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-429ee328c2642962","role":"sequence","name":"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:  0;\n   2:  0;\n   3:  2;\n   4:  8,   6;\n   5: 16,  36,   12;\n   6: 24, 108,   96,   20;\n   7: 32, 228,  384,  200,   30;\n   8: 40, 396, 1056, 1000,  360,   42;\n   9: 48, 612, 2304, 3400, 2160,  588,  56;\n  10: 56, 876, 4320, 9000, 8760, 4116, 896, 72;\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) = 0, P(2,x) = 0, P(3,x) = 2, P(4,x) = 8 + 6*x, P(5,x) = 16 + 36*x + 12*x^2, P(6,x) = 24 + 108*x + 96*x^2 + 20*x^3.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 2;\nP[4, x_] := 8 + 6*x;\nP[5, x_] := 16 + 36*x + 12*x^2;\nP[6, x_] := 24 + 108*x + 96*x^2 + 20*x^3;\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] = 0; v[2] = 0; v[3] = 2; v[4] = 8 + 6*x; v[5] = 16 + 36*x + 12*x^2; v[6] = 24 + 108*x + 96*x^2 + 20*x^3; 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","D"],"edges":["e-31b80a123240a0c6","e-fe5780763608ad2e"],"text":"A122542 \u2192 D \u2192 D"},"headline":"A122542 \u2192 D \u2192 D","chain":null}