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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":"94f72b9198c32e8745c5f56d1b40a98cc89ebfbd2c4f71ca8d3685891503ffbf","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-f8f37b87137bc296","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","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:   2;\n  3:   4;\n  4:  14,    8;\n  5:  32,   24;\n  6:  58,   96,   24;\n  7:  92,  272,   96;\n  8: 134,  600,  432,  64;\n  9: 184, 1128, 1440, 320;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-17*n^2+48*n-44+(8*n^2-51*n+70)*x+(8*n-16)*x^2)*P(n,x) = (2*n^3-13*n^2+29*n-22+(8*n^3-68*n^2+189*n-193)*x+(32*n^2-196*n+316)*x^2+(32*n-96)*x^3)*P(n-1,x) - ((2*n^2-9*n+11)*x+(16*n^2-64*n+69)*x^2+(64*n-144)*x^3+64*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 13*n^2 + 29*n - 22 + (8*n^3 - 68*n^2 + 189*n - 193)*x + (32*n^2 - 196*n + 316)*x^2 + (32*n - 96)*x^3)*P[n - 1, x] - ((2*n^2 - 9*n + 11)*x + (16*n^2 - 64*n + 69)*x^2 + (64*n - 144)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(2*n^3 - 17*n^2 + 48*n - 44 + (8*n^2 - 51*n + 70)*x + (8*n - 16)*x^2)]];\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] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^3 - 13*n^2 + 29*n - 22 + (8*n^3 - 68*n^2 + 189*n - 193)*x + (32*n^2 - 196*n + 316)*x^2 + (32*n - 96)*x^3)*v[i - 1] - ((2*n^2 - 9*n + 11)*x + (16*n^2 - 64*n + 69)*x^2 + (64*n - 144)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(2*n^3 - 17*n^2 + 48*n - 44 + (8*n^2 - 51*n + 70)*x + (8*n - 16)*x^2)); 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-e3d901814302c059","steps":["D","D+1"],"edges":["e-3a54b8685030d0ee","e-1bdf131250a78a6a"],"text":"A128099 \u2192 D \u2192 D+1"},"headline":"A128099 \u2192 D \u2192 D+1","chain":null}