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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":"a30b4f7bff2b4d962d9e0fca54b9ae9bce3287291ee741b51f4e99e75bed62c7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-c1c65e1c83bcd2f2","role":"sequence","name":"Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0,  -1;\n   3: 0,  -3, -1;\n   4: 0,  -5,  1,  1;\n   5: 0,  -7,  3,  6,   1;\n   6: 0,  -9,  5, 15,  -2,   -1;\n   7: 0, -11,  7, 28,  -9,   -9, -1;\n   8: 0, -13,  9, 45, -20,  -30,  3,  1;\n   9: 0, -15, 11, 66, -35,  -70, 18, 12,  1;\n  10: 0, -17, 13, 91, -54, -135, 50, 50, -4, -1;\n","recurrence_sentence":"The row polynomials satisfy (8*n^3-52*n^2+110*n-75+(16*n^2-72*n+72)*x+(8*n-12)*x^2)*P(n,x) = (8*n^3-40*n^2+66*n-36+(12*n^2-44*n+39)*x-(16*n^3-96*n^2+180*n-112)*x^2-(32*n^2-128*n+132)*x^3-(16*n-32)*x^4)*P(n-1,x) - ((4*n^2-12*n+9)*x+(8*n-12)*x^2-(16*n^2-48*n+32)*x^3-(32*n-48)*x^4-16*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 - 40*n^2 + 66*n - 36 + (12*n^2 - 44*n + 39)*x - (16*n^3 - 96*n^2 + 180*n - 112)*x^2 - (32*n^2 - 128*n + 132)*x^3 - (16*n - 32)*x^4)*P[n - 1, x] - ((4*n^2 - 12*n + 9)*x + (8*n - 12)*x^2 - (16*n^2 - 48*n + 32)*x^3 - (32*n - 48)*x^4 - 16*x^5)*D[P[n - 1, x], x])/(8*n^3 - 52*n^2 + 110*n - 75 + (16*n^2 - 72*n + 72)*x + (8*n - 12)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((8*n^3 - 40*n^2 + 66*n - 36 + (12*n^2 - 44*n + 39)*x - (16*n^3 - 96*n^2 + 180*n - 112)*x^2 - (32*n^2 - 128*n + 132)*x^3 - (16*n - 32)*x^4)*v[i - 1] - ((4*n^2 - 12*n + 9)*x + (8*n - 12)*x^2 - (16*n^2 - 48*n + 32)*x^3 - (32*n - 48)*x^4 - 16*x^5)*deriv(v[i - 1], x))/(8*n^3 - 52*n^2 + 110*n - 75 + (16*n^2 - 72*n + 72)*x + (8*n - 12)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-c1c65e1c83bcd2f2","steps":["D+1"],"edges":["e-b113ea8f5eac0e8f"],"text":"A108299 \u2192 D+1"},"headline":"A108299 \u2192 D+1","chain":null}