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for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"4ff02ec9bc94052c0075832af5e603082518ec7f7666d4c98daed12dd8b2ebb1","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: -1,  -3,  -1;\n   4: -1,  -2,   1,   1;\n   5:  0,   3,   9,   6,   1;\n   6:  1,   7,  13,   5,  -2,  -1;\n   7:  1,   4,  -2, -22, -24,  -9,  -1;\n   8:  0,  -5, -30, -60, -45,  -9,   3,  1;\n   9: -1, -11, -37, -38,  25,  70,  46, 12,  1;\n  10: -1,  -6,   3,  87, 220, 229, 106, 14, -4, -1;\n","recurrence_sentence":"The row polynomials satisfy (2*n-3)*P(n,x) = -(2*n-4+(8*n-15)*x+(4*n-8)*x^2)*P(n-1,x) + (3+11*x+12*x^2+4*x^3)*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[(-(2*n - 4 + (8*n - 15)*x + (4*n - 8)*x^2)*P[n - 1, x] + (3 + 11*x + 12*x^2 + 4*x^3)*D[P[n - 1, x], x])/(2*n - 3)]];\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] = (-(2*n - 4 + (8*n - 15)*x + (4*n - 8)*x^2)*v[i - 1] + (3 + 11*x + 12*x^2 + 4*x^3)*deriv(v[i - 1], x))/(2*n - 3)); 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":["x \u21a6 x+1"],"edges":["e-b80b7af82ad2cf71"],"text":"A108299 \u2192 x \u21a6 x+1"},"headline":"A108299 \u2192 x \u21a6 x+1","chain":null}