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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"70725e61c13ecc907c4adab33b33325bf35039da266dee5c438d71f786245391","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-de2147e0b2be5082","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":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:        9,        24;\n  2:       53,       252,        288;\n  3:      309,      2124,       4752,        3456;\n  4:     1801,     16200,      53712,       77760,       41472;\n  5:    10497,    116688,     511920,     1107648,     1181952,      497664;\n  6:    61181,    809892,    4418064,    12711168,    20342016,    17169408,     5971968;\n  7:   356589,   5476836,   35715168,   128176128,   273404160,   346622976,   241864704,    71663616;\n  8:  2078353,  36330192,  275594976,  1184927616,  3158196480,  5343418368,  5604691968,  3332358144,   859963392;\n  9: 12113529, 237444552, 2053816992, 10288529280, 32894906112, 69612244992, 97507307520, 87178788864, 45148078080, 10319560704;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 24*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 9 + 24*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (6 + 12*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (6 + 12*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-353fb5a029e4e669","steps":["basis A013610","basis A013611"],"edges":["e-4d45de18adc3c05d","e-9870ed7f533cff1c"],"text":"A207607 \u2192 basis A013610 \u2192 basis A013611"},"headline":"A207607 \u2192 basis A013610 \u2192 basis A013611","chain":null}