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(elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"f7a861de852fa4084d40fcf55a69d9ff33a5c2add78693891584bd9b4a08600f","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-d5dd236ae5fa65ba","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","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, 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488477696,   167014400,    36569088,    4653056,    262144;\n  8:  811826071,  4239499180,  9928349424,  13730696448,  12418651392,   7675358208,  3282944000,   959578112,  183435264,  20709376,  1048576;\n  9: 6391493137, 36678059304, 95373851536, 148335108288, 153323201280, 110588993536, 56797970432, 20771831808, 5301141504, 899153920, 91226112, 4194304;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (8+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 55 + 60*x + 16*x^2, P(1,x) = 433 + 696*x + 368*x^2 + 64*x^3.","mathematica":"P[0, x_] := 55 + 60*x + 16*x^2;\nP[1, x_] := 433 + 696*x + 368*x^2 + 64*x^3;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (8 + 4*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] = 55 + 60*x + 16*x^2; v[2] = 433 + 696*x + 368*x^2 + 64*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (8 + 4*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-8eb9bb3a09f0b7fd","steps":["basis A013609","basis A007318"],"edges":["e-125a67b606fd2347","e-1233760a867099ea"],"text":"A123519 \u2192 basis A013609 \u2192 basis A007318"},"headline":"A123519 \u2192 basis A013609 \u2192 basis A007318","chain":null}