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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"d98375d04cd36f6d63fdf105765ee5f90ff8fb8d9e4d65a07b3d2b1f17cf4feb","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-9c403911703f1b61","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 2, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.","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:       4,       12;\n   3:      24,      120,       144;\n   4:     136,      984,      2304,       1728;\n   5:     776,     7320,     25344,      38016,       20736;\n   6:    4424,    51480,    235584,     528768,      580608,      248832;\n   7:   25224,   349080,   1984896,    5927040,     9787392,     8460288,     2985984;\n   8:  143816,  2307096,  15673536,   58396032,   128729088,   167712768,   119439360,    35831808;\n   9:  819976, 14960280, 118181376,  527589504,  1454671872,  2534602752,  2723217408,  1648263168,   429981696;\n  10: 4675144, 95595288, 860809536, 4477790592, 14820102144, 32344676352, 46521630720, 42496524288, 22359048192, 5159780352;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+12*x)*P(n-2,x) + (5+12*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4 + 12*x;\nP[n_, x_] := P[n, x] = Expand[(4 + 12*x)*P[n - 2, x] + (5 + 12*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] = 4 + 12*x; for(n = 3, N, my(i = n); v[i] = (4 + 12*x)*v[i - 2] + (5 + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b0fc620949fb5990","steps":["basis A013610","basis A013611"],"edges":["e-7f7d545e75881e09","e-65acb93fa25ec3a3"],"text":"A122542 \u2192 basis A013610 \u2192 basis A013611"},"headline":"A122542 \u2192 basis A013610 \u2192 basis A013611","chain":null}