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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":"77ea0aa9e08b1475e1be4c1d9821a5777448423765d63c30f3a0fd36e70261a6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0b318b4ec2091cb2","role":"sequence","name":"Triangle of coefficients of Gaussian polynomials [2n+1,1]_q represented as finite sum of terms (1+q^2)^k*q^(g-k), where k = 0,1,...,g with g=n.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"}]}],"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:  2,   1;\n  2:  0,   3,   1;\n  3: -3,   0,   4,   1;\n  4: -1,  -8,   0,   5,   1;\n  5:  4,  -3, -15,   0,   6,   1;\n  6:  2,  15,  -6, -24,   0,   7,   1;\n  7: -5,   8,  36, -10, -35,   0,   8, 1;\n  8: -3, -24,  20,  70, -15, -48,   0, 9,  1;\n  9:  6, -15, -70,  40, 120, -21, -63, 0, 10, 1;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-3*n^2+13*n-6+(4*n-2)*x-(2*n-1)*x^2)*P(n,x) = (n^2+5*n+6+(n^3+n^2+6*n+2)*x+(n-1)*x^2-n*x^3)*P(n-1,x) - (4*n^2+24-(n^2+10)*x^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[((n^2 + 5*n + 6 + (n^3 + n^2 + 6*n + 2)*x + (n - 1)*x^2 - n*x^3)*P[n - 1, x] - (4*n^2 + 24 - (n^2 + 10)*x^2 + x^4)*D[P[n - 1, x], x])/(2*n^3 - 3*n^2 + 13*n - 6 + (4*n - 2)*x - (2*n - 1)*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] = ((n^2 + 5*n + 6 + (n^3 + n^2 + 6*n + 2)*x + (n - 1)*x^2 - n*x^3)*v[i - 1] - (4*n^2 + 24 - (n^2 + 10)*x^2 + x^4)*deriv(v[i - 1], x))/(2*n^3 - 3*n^2 + 13*n - 6 + (4*n - 2)*x - (2*n - 1)*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-0b318b4ec2091cb2","steps":["D+1"],"edges":["e-50c0ba76f50a7e5c"],"text":"A267482 \u2192 D+1"},"headline":"A267482 \u2192 D+1","chain":null}