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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":"ae201fbec8003a71a6ce4e95343f7f60d1a46aabc34efceffad0b4d61c83a9d0","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8217f7f3de935aeb","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.","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:       7,        4;\n  2:      41,       70,        20;\n  3:     231,      690,       588,        112;\n  4:    1289,     5572,      8288,       4704,        672;\n  5:    7183,    40840,     87840,      86880,      36960,       4224;\n  6:   40081,   283122,    800820,    1144440,     847440,     288288,      27456;\n  7:  224143,  1894270,   6657420,   12507880,   13373360,    7911904,    2242240,    183040;\n  8: 1256465, 12369544,  52022432,  121433312,  170634464,  145681536,   71751680,  17425408,   1244672;\n  9: 7059735, 79369164, 388937952, 1086470112, 1896489504, 2127250944, 1511258112, 637498368, 135442944, 8599552;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+6*n^2+5*n+2+(5*n+10)*x-(2*n+4)*x^2)*P(n,x) = (6*n^3+30*n^2+47*n+22+(8*n^3+40*n^2+71*n+46)*x+(6*n+12)*x^2-(8*n+16)*x^3)*P(n-1,x) + (4*n^2+8*n+4+(12*n^2+24*n+14)*x+(8*n^2+16*n+10)*x^2-8*x^3-8*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[((6*n^3 + 30*n^2 + 47*n + 22 + (8*n^3 + 40*n^2 + 71*n + 46)*x + (6*n + 12)*x^2 - (8*n + 16)*x^3)*P[n - 1, x] + (4*n^2 + 8*n + 4 + (12*n^2 + 24*n + 14)*x + (8*n^2 + 16*n + 10)*x^2 - 8*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 + 6*n^2 + 5*n + 2 + (5*n + 10)*x - (2*n + 4)*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] = ((6*n^3 + 30*n^2 + 47*n + 22 + (8*n^3 + 40*n^2 + 71*n + 46)*x + (6*n + 12)*x^2 - (8*n + 16)*x^3)*v[i - 1] + (4*n^2 + 8*n + 4 + (12*n^2 + 24*n + 14)*x + (8*n^2 + 16*n + 10)*x^2 - 8*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 + 6*n^2 + 5*n + 2 + (5*n + 10)*x - (2*n + 4)*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-66e2c3448245b0b7","steps":["basis A013609","D+1"],"edges":["e-4e2df72103232439","e-d09d30b6c33ac327"],"text":"A033282 \u2192 basis A013609 \u2192 D+1"},"headline":"A033282 \u2192 basis A013609 \u2192 D+1","chain":null}