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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":"30a2b0b7db32ec4cd1686396d59fd25fafdbebb9747f14591b887fd5f2b7629f","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-7bf0ce499426260c","role":"sequence","name":"derivative of basis-map coefficients of Triangular array of generalized Narayana numbers: T(n, k) = 2*binomial(n + 1, k + 1)*binomial(n + 1, k - 1)/(n + 1). shifted by x ->  x + 1","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:    0;\n  1:    0;\n  2:    6;\n  3:   40,    24;\n  4:  150,   240,     60;\n  5:  420,  1260,    840,    120;\n  6:  980,  4704,   5880,   2240,    210;\n  7: 2016, 14112,  28224,  20160,   5040,    336;\n  8: 3780, 36288, 105840, 120960,  56700,  10080,   504;\n  9: 6600, 83160, 332640, 554400, 415800, 138600, 18480, 720;\n","recurrence_sentence":"The row polynomials satisfy (n^3+2*n^2-8*n)*P(n,x) = (n^3+7*n^2+14*n+8+(3*n^3+3*n^2-6*n-6)*x)*P(n-1,x) + ((2*n^2+4*n+2)*x-(2*n^2+4*n+2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 + 7*n^2 + 14*n + 8 + (3*n^3 + 3*n^2 - 6*n - 6)*x)*P[n - 1, x] + ((2*n^2 + 4*n + 2)*x - (2*n^2 + 4*n + 2)*x^2)*D[P[n - 1, x], x])/(n^3 + 2*n^2 - 8*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 6; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 + 7*n^2 + 14*n + 8 + (3*n^3 + 3*n^2 - 6*n - 6)*x)*v[i - 1] + ((2*n^2 + 4*n + 2)*x - (2*n^2 + 4*n + 2)*x^2)*deriv(v[i - 1], x))/(n^3 + 2*n^2 - 8*n)); 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-0e6d847331141ca6","steps":["D","D"],"edges":["e-e77b91d6fbb51ebc","e-2bd3ff0b5f26fc6c"],"text":"A145596 \u2192 D \u2192 D"},"headline":"A145596 \u2192 D \u2192 D","chain":null}