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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"2928b5fb41138dbe0629fe79afe5abe9d8a52c0524d57813917f87e2a87a3cf7","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-b0fc620949fb5990","role":"sequence","name":"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:    1,     1;\n   3:    3,     5,      1;\n   4:    7,    21,     10,      1;\n   5:   17,    83,     69,     16,      1;\n   6:   41,   315,    410,    163,     23,     1;\n   7:   99,  1169,   2245,   1358,    322,    31,     1;\n   8:  239,  4277,  11694,  10097,   3568,   568,    40,    1;\n   9:  577, 15507,  58957,  69856,  34473,  8052,   926,   50,  1;\n  10: 1393, 55891, 290694, 460319, 304631, 97813, 16308, 1424, 61, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-2,x) + x*d/dx P(n-2,x) + (2+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 2, x] + x*D[P[n - 2, x], x] + (2 + x)*P[n - 1, x] + 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] = 1 + x; for(n = 3, N, my(i = n); v[i] = (1 + x)*v[i - 2] + x*deriv(v[i - 2], x) + (2 + x)*v[i - 1] + 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 A008277"],"edges":["e-f55b565f0f484977"],"text":"A122542 \u2192 basis A008277"},"headline":"A122542 \u2192 basis A008277","chain":null}