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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"7a4f97cc5006e6a4a3c6769aa0708de34eae0c0bc70908dd584e087fcc7de9e2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-84dfb3e427b94d98","role":"sequence","name":"basis_map_A013609 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:      3,        2;\n   3:     15,       20,         4;\n   4:     69,      166,        76,         8;\n   5:    321,     1264,      1008,       240,        16;\n   6:   1491,     9194,     11456,      4640,       688,       32;\n   7:   6927,    65108,    119620,     73488,     18192,     1856,      64;\n   8:  32181,   453454,   1185756,   1038792,    382128,    64160,    4800,    128;\n   9: 149505,  3124480,  11360512,  13660064,   7010208,  1723008,  209792,  12032,   256;\n  10: 694563, 21377522, 106340256, 171042304, 117723616, 39740864, 7003008, 647936, 29440, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+2*x)*P(n-2,x) + 2*x*d/dx P(n-2,x) + (4+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + (4 + 2*x)*P[n - 1, x] + 2*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = (3 + 2*x)*v[i - 2] + 2*x*deriv(v[i - 2], x) + (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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-b0fc620949fb5990","steps":["basis A013609","basis A008277"],"edges":["e-778f192f6d6aff27","e-23882e9f4ba0be93"],"text":"A122542 \u2192 basis A013609 \u2192 basis A008277"},"headline":"A122542 \u2192 basis A013609 \u2192 basis A008277","chain":null}