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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"e1e768ee1c659497ba54d405d466172a382690a7bb54ffa38700e92c3a602e90","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-831045384d9c95c9","role":"sequence","name":"basis_map_A013610 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:         5,         12;\n   3:        41,        168,         144;\n   4:       365,       2196,        3888,         1728;\n   5:      3281,      26256,       72288,        76032,        20736;\n   6:     29525,     295260,     1107360,      1883520,      1347840,       248832;\n   7:    265721,    3188664,    15146352,     35741952,     42612480,     22394880,      2985984;\n   8:   2391485,   33480804,   192514896,    581732928,    976727808,    876137472,    355332096,     35831808;\n   9:  21523361,  344373792,  2324523456,   8566304256,  18590860800,  23786348544,  16805117952,   5446434816,   429981696;\n  10: 193710245, 3486784428, 27022579776, 117533700864, 313102444032, 522064963584, 531994853376, 305716985856, 81266540544, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9+72*x+144*x^2)*P(n-2,x) + (10+24*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 5 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-(9 + 72*x + 144*x^2)*P[n - 2, x] + (10 + 24*x)*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] = 5 + 12*x; for(n = 3, N, my(i = n); v[i] = -(9 + 72*x + 144*x^2)*v[i - 2] + (10 + 24*x)*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":["basis A013610","basis A013611"],"edges":["e-3b5bae8e6d048f8c","e-83c5bd15f7c9704f"],"text":"A086645 \u2192 basis A013610 \u2192 basis A013611"},"headline":"A086645 \u2192 basis A013610 \u2192 basis A013611","chain":null}