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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"9bf62997b6abbe15d28c96c59db578deedde375414c9b284fb38df3ef08b6b87","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-35273d9764d27bc4","role":"sequence","name":"basis_map_A013611 of Triangular array formed by the Mersenne numbers.","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:         8,          4;\n  2:        62,         80,          16;\n  3:       480,       1200,         576,          64;\n  4:      3716,      16160,       13984,        3584,         256;\n  5:     28768,     206384,      287232,      127488,       20480,        1024;\n  6:    222712,    2559360,     5393280,     3691520,     1000960,      110592,       4096;\n  7:   1724160,   31190400,    95955456,    95148544,    38748160,     7098368,     573440,     16384;\n  8:  13347856,  376062720,  1649262336,  2279409664,  1308548096,   353525760,   46735360,   2883584,    65536;\n  9: 103334528, 4503763264, 27700537344, 51994945536, 40445296640, 15118716928, 2908487680, 290717696, 14155776, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -2*P(n-2,x) + (8+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 8 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 8 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-2*P[n - 2, x] + (8 + 4*x)*P[n - 1, x] + 4*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] = 1; v[2] = 8 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -2*v[i - 2] + (8 + 4*x)*v[i - 1] + 4*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-9b5c7e97e5191da6","steps":["basis A013611","basis A008277"],"edges":["e-6b7a7c8afbb896b5","e-618fb32af439fcc5"],"text":"A110441 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A110441 \u2192 basis A013611 \u2192 basis A008277","chain":null}