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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":"7a3329b6a34ae2e70a272597bf5989641a4581925d5224ad25215e2c70de3ebe","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-4479b0484a11b442","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ternary trees with n edges and having k middle edges (n >= 0, k >= 0).","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:      3,      1;\n   3:     11,      8,       1;\n   4:     42,     52,      15,       1;\n   5:    162,    300,     150,      24,      1;\n   6:    627,   1595,    1210,     340,     35,      1;\n   7:   2431,   8008,    8463,    3640,    665,     48,     1;\n   8:   9438,  38584,   53508,   32340,   9100,   1176,    63,    1;\n   9:  36686, 180336,  314160,  251328,  99960,  19992,  1932,   80,  1;\n  10: 142766, 823650, 1744200, 1767456, 936054, 265734, 39900, 3000, 99, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-6*n^3+8*n^2+6*n+(8*n^3-16*n^2-10*n)*x+(4*n^2+2*n)*x^2)*P(n,x) = (15*n^4-22*n^3+43*n^2-32*n+8+(12*n^4-2*n^3-65*n^2+47*n-10)*x+(24*n^3-65*n^2+61*n-14)*x^2+(12*n^2-16*n+4)*x^3)*P(n-1,x) + (n^3-3*n^2+5*n-3+(7*n^3-n^2-7*n+7)*x-(8*n^3-20*n^2+25*n-1)*x^2-(16*n^2-35*n+7)*x^3-(8*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((15*n^4 - 22*n^3 + 43*n^2 - 32*n + 8 + (12*n^4 - 2*n^3 - 65*n^2 + 47*n - 10)*x + (24*n^3 - 65*n^2 + 61*n - 14)*x^2 + (12*n^2 - 16*n + 4)*x^3)*P[n - 1, x] + (n^3 - 3*n^2 + 5*n - 3 + (7*n^3 - n^2 - 7*n + 7)*x - (8*n^3 - 20*n^2 + 25*n - 1)*x^2 - (16*n^2 - 35*n + 7)*x^3 - (8*n - 2)*x^4)*D[P[n - 1, x], x])/(4*n^4 - 6*n^3 + 8*n^2 + 6*n + (8*n^3 - 16*n^2 - 10*n)*x + (4*n^2 + 2*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((15*n^4 - 22*n^3 + 43*n^2 - 32*n + 8 + (12*n^4 - 2*n^3 - 65*n^2 + 47*n - 10)*x + (24*n^3 - 65*n^2 + 61*n - 14)*x^2 + (12*n^2 - 16*n + 4)*x^3)*v[i - 1] + (n^3 - 3*n^2 + 5*n - 3 + (7*n^3 - n^2 - 7*n + 7)*x - (8*n^3 - 20*n^2 + 25*n - 1)*x^2 - (16*n^2 - 35*n + 7)*x^3 - (8*n - 2)*x^4)*deriv(v[i - 1], x))/(4*n^4 - 6*n^3 + 8*n^2 + 6*n + (8*n^3 - 16*n^2 - 10*n)*x + (4*n^2 + 2*n)*x^2)); 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-4479b0484a11b442","steps":["D+1"],"edges":["e-9706c5f033ea0419"],"text":"A120986 \u2192 D+1"},"headline":"A120986 \u2192 D+1","chain":null}