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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":"2f650894171bb8afa4bc4be6fde61d2dc36826187cb77cdc0116848955efa933","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-83e32d0bc1c10d45","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-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:       1;\n  1:       4,       1;\n  2:      18,       9,       1;\n  3:      87,      66,      15,       1;\n  4:     439,     450,     158,      22,      1;\n  5:    2278,    2955,    1420,     310,     30,      1;\n  6:   12052,   18963,   11640,    3515,    540,     39,     1;\n  7:   64669,  119812,   89838,   34930,   7490,    868,    49,    1;\n  8:  350733,  748548,  664552,  317576,  88284,  14392,  1316,   60,  1;\n  9: 1918152, 4637205, 4763412, 2708328, 930384, 197820, 25620, 1908, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^3+5*n^2+6*n+(2*n^2+5*n+2)*x+(n+2)*x^2)*P(n,x) = (7*n^3+29*n^2+12*n+(3*n^3+25*n^2+18*n+2)*x+(6*n^2+13*n+2)*x^2+3*n*x^3)*P(n-1,x) - (4*n^2+18*n+12+(6*n^2+35*n+30)*x+(2*n^2+21*n+28)*x^2+(4*n+12)*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[((7*n^3 + 29*n^2 + 12*n + (3*n^3 + 25*n^2 + 18*n + 2)*x + (6*n^2 + 13*n + 2)*x^2 + 3*n*x^3)*P[n - 1, x] - (4*n^2 + 18*n + 12 + (6*n^2 + 35*n + 30)*x + (2*n^2 + 21*n + 28)*x^2 + (4*n + 12)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^3 + 5*n^2 + 6*n + (2*n^2 + 5*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] = ((7*n^3 + 29*n^2 + 12*n + (3*n^3 + 25*n^2 + 18*n + 2)*x + (6*n^2 + 13*n + 2)*x^2 + 3*n*x^3)*v[i - 1] - (4*n^2 + 18*n + 12 + (6*n^2 + 35*n + 30)*x + (2*n^2 + 21*n + 28)*x^2 + (4*n + 12)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^3 + 5*n^2 + 6*n + (2*n^2 + 5*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-57b4b14604f4c0d7","steps":["basis A007318","D+1"],"edges":["e-3c8d43f1dfa996d6","e-76f545b673d694d0"],"text":"A126216 \u2192 basis A007318 \u2192 D+1"},"headline":"A126216 \u2192 basis A007318 \u2192 D+1","chain":null}