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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":"675d523fec73b6cd386912db8264beb40759fb8cb6ac618363124b041bfe3082","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-af7a69a897cce4e9","role":"sequence","name":"basis_map_A013611 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:    1;\n  1:    1;\n  2:   11,     8;\n  3:   21,    16;\n  4:   67,   184,     64;\n  5:  149,   512,    192;\n  6:  371,  1864,   2304,    512;\n  7:  837,  5104,   8320,   2048;\n  8: 1923, 14424,  33984,  25600,  4096;\n  9: 4277, 37344, 108864, 112640, 20480;\n","recurrence_sentence":"The row polynomials satisfy (128*n^3-240*n^2-44*n+(512*n^2-1008*n)*x+512*n*x^2)*P(n,x) = (640*n^3-1584*n^2+164*n+624+(2048*n^3-3328*n^2-3696*n+4480)*x+(8192*n^2-13568*n+5888)*x^2+(8192*n-8192)*x^3)*P(n-1,x) - (288*n^2+36*n-63+(2176*n^2+1424*n-440)*x+(4096*n^2+9216*n+528)*x^2+(16384*n+9216)*x^3+16384*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((640*n^3 - 1584*n^2 + 164*n + 624 + (2048*n^3 - 3328*n^2 - 3696*n + 4480)*x + (8192*n^2 - 13568*n + 5888)*x^2 + (8192*n - 8192)*x^3)*P[n - 1, x] - (288*n^2 + 36*n - 63 + (2176*n^2 + 1424*n - 440)*x + (4096*n^2 + 9216*n + 528)*x^2 + (16384*n + 9216)*x^3 + 16384*x^4)*D[P[n - 1, x], x])/(128*n^3 - 240*n^2 - 44*n + (512*n^2 - 1008*n)*x + 512*n*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((640*n^3 - 1584*n^2 + 164*n + 624 + (2048*n^3 - 3328*n^2 - 3696*n + 4480)*x + (8192*n^2 - 13568*n + 5888)*x^2 + (8192*n - 8192)*x^3)*v[i - 1] - (288*n^2 + 36*n - 63 + (2176*n^2 + 1424*n - 440)*x + (4096*n^2 + 9216*n + 528)*x^2 + (16384*n + 9216)*x^3 + 16384*x^4)*deriv(v[i - 1], x))/(128*n^3 - 240*n^2 - 44*n + (512*n^2 - 1008*n)*x + 512*n*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":["basis A013611","D+1"],"edges":["e-099e290d18ba9894","e-173d41b78f81748f"],"text":"A128099 \u2192 basis A013611 \u2192 D+1"},"headline":"A128099 \u2192 basis A013611 \u2192 D+1","chain":null}