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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":"03a5dcb88967b408afa4492622f5e1d6f9107287da488a3591cdb99c49435cbc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-948c4fae35fcbf15","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":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:     1;\n   3:     2,      1;\n   4:    16,     15;\n   5:    72,     72,      1;\n   6:   256,    300,     45;\n   7:   992,   1488,    498,     1;\n   8:  4096,   7280,   3276,    91;\n   9: 16512,  33024,  18336,  1824,   1;\n  10: 65536, 146880, 100368, 19176, 153;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) + (16+12*x)*P(n-3,x) - (4-2*x)*P(n-2,x) + 4*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1, P(3,x) = 2 + x, P(4,x) = 16 + 15*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 2 + x;\nP[4, x_] := 16 + 15*x;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] + (16 + 12*x)*P[n - 3, x] - (4 - 2*x)*P[n - 2, x] + 4*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1; v[3] = 2 + x; v[4] = 16 + 15*x; for(n = 5, N, my(i = n); v[i] = -x^2*v[i - 4] + (16 + 12*x)*v[i - 3] - (4 - 2*x)*v[i - 2] + 4*v[i - 1]); 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-f7935c36a3c0ec8c","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-d4161fce9e8d4a19","e-003dc54f0db56643"],"text":"A086645 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A086645 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}