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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"7a6404a0609aa9af4c9e2ada5d98b734c3f1c2d1ca9a354d8d29c81fa97b6197","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-8eb9bb3a09f0b7fd","role":"sequence","name":"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, 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:      11,        18,          4;\n  1:      41,       128,         68,           8;\n  2:     153,       832,        796,         216,          16;\n  3:     571,      5170,       7964,        3744,         624,         32;\n  4:    2131,     31330,      73256,       53152,       14960,       1696,         64;\n  5:    7953,    187072,     640744,      674288,      285200,      53632,       4416,       128;\n  6:   29681,   1107008,    5426840,     7971216,     4756016,    1319552,     177856,     11136,      256;\n  7:  110771,   6514338,   44987992,    89891552,    72729424,   27932128,    5480384,    556032,    27392,     512;\n  8:  413403,  38200562,  367505772,   980920288,  1047780176,  535189088,  143372544,  20958208,  1659648,   66048,   1024;\n  9: 1542841, 223515840, 2971459308, 10454322872, 14462473264, 9560275456, 3358858496, 663436800, 75081984, 4771840, 156672, 2048;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 11 + 18*x + 4*x^2, P(1,x) = 41 + 128*x + 68*x^2 + 8*x^3.","mathematica":"P[0, x_] := 11 + 18*x + 4*x^2;\nP[1, x_] := 41 + 128*x + 68*x^2 + 8*x^3;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 11 + 18*x + 4*x^2; v[2] = 41 + 128*x + 68*x^2 + 8*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 2*x)*v[i - 1] + 2*x*deriv(v[i - 1], x)); 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 A008277"],"edges":["e-ad85cdc8ea03f732"],"text":"A123519 \u2192 basis A008277"},"headline":"A123519 \u2192 basis A008277","chain":null}