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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":"532fdf8e0228af77b4f3c70be282c1eb1b1643b78542d09c1f45d0687e0ee1ba","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-07af2e6e3c9f0c77","role":"sequence","name":"basis_map_A007318 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:       5,        2;\n  2:      26,       25,        5;\n  3:     138,      219,      105,       14;\n  4:     743,     1666,     1316,      420,       42;\n  5:    4043,    11780,    13140,     6900,     1650,      132;\n  6:   22180,    79731,   115560,    85635,    33660,     6435,      429;\n  7:  122468,   524405,   937860,   902825,   500500,   157157,    25025,    1430;\n  8:  679757,  3380740,  7203812,  8556548,  6160154,  2726724,   712712,   97240,   4862;\n  9: 3789297, 21477510, 53169012, 75248628, 66835314, 38407824, 14145768, 3166488, 377910, 16796;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n^2+4*n+4+(n+2)*x-(n+2)*x^2)*P(n,x) = (3*n^3+15*n^2+26*n+16+(2*n^3+10*n^2+17*n+10)*x-(3*n+6)*x^2-(2*n+4)*x^3)*P(n-1,x) + (4*n^2+8*n+4+(6*n^2+12*n+2)*x+(2*n^2+4*n-8)*x^2-8*x^3-2*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[((3*n^3 + 15*n^2 + 26*n + 16 + (2*n^3 + 10*n^2 + 17*n + 10)*x - (3*n + 6)*x^2 - (2*n + 4)*x^3)*P[n - 1, x] + (4*n^2 + 8*n + 4 + (6*n^2 + 12*n + 2)*x + (2*n^2 + 4*n - 8)*x^2 - 8*x^3 - 2*x^4)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 4*n + 4 + (n + 2)*x - (n + 2)*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] = ((3*n^3 + 15*n^2 + 26*n + 16 + (2*n^3 + 10*n^2 + 17*n + 10)*x - (3*n + 6)*x^2 - (2*n + 4)*x^3)*v[i - 1] + (4*n^2 + 8*n + 4 + (6*n^2 + 12*n + 2)*x + (2*n^2 + 4*n - 8)*x^2 - 8*x^3 - 2*x^4)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 4*n + 4 + (n + 2)*x - (n + 2)*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":["x \u21a6 x+1","D+1"],"edges":["e-190c283dd8b47413","e-242427e1bd4a160c"],"text":"A033282 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A033282 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}