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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"500a0fb38d659b33a9846f00b668649fc6d09b03b352d8e3d9590cab77923075","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7d332fcb7bf9a0f7","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of ways to tile a 2 X n rectangle with k pieces of 2 X 2 tiles and n-2k pieces of 1 X 2 tiles (0 <= k <= floor(n/2)).","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:   1;\n  2:   3,   1;\n  3:   5,   2;\n  4:  11,   8,   1;\n  5:  21,  19,   3;\n  6:  43,  54,  15,  1;\n  7:  85, 132,  46,  4;\n  8: 171, 337, 160, 24, 1;\n  9: 341, 818, 476, 90, 5;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+x)*P(n-2,x) + x*d/dx P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2 + x)*P[n - 2, x] + x*D[P[n - 2, x], 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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (2 + x)*v[i - 2] + x*deriv(v[i - 2], 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-75ccd0453a9b6cec","steps":["basis A007318","basis A008277"],"edges":["e-2a29a8b9938e2bd7","e-68bbb0ff992bfa4f"],"text":"A011973 \u2192 basis A007318 \u2192 basis A008277"},"headline":"A011973 \u2192 basis A007318 \u2192 basis A008277","chain":null}