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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"c352e5a94ce415336d2941ebbb17cd7a30d53a46dc4bf7ee7a33bb18b2a78fd2","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-ea2139b8dbf6bf8d","role":"sequence","name":"Interpolation operator: Triangle with an even number of zeros in each line followed by 1 or 2 ones.","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: 2,   1;\n  2: 1,   3,     1;\n  3: 2,  10,     7,     1;\n  4: 1,  15,    25,    10,     1;\n  5: 2,  46,   115,    75,    16,     1;\n  6: 1,  63,   301,   350,   140,    21,    1;\n  7: 2, 190,  1267,  2051,  1190,   287,   29,   1;\n  8: 1, 255,  3025,  7770,  6951,  2646,  462,  36,  1;\n  9: 2, 766, 12355, 41875, 49476, 25473, 6342, 786, 46, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x+x^2)*P(n-2,x) + (3*x+2*x^2)*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*x + x^2)*P[n - 2, x] + (3*x + 2*x^2)*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + 3*x + x^2)*v[i - 2] + (3*x + 2*x^2)*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), 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-ea2139b8dbf6bf8d","steps":["basis A008277"],"edges":["e-5d454ac027f458ad"],"text":"A133080 \u2192 basis A008277"},"headline":"A133080 \u2192 basis A008277","chain":null}