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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"61060cd4110d3819ed4dbb6703a737c7b55a900384d06df8da737ce62c721f71","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-e7c079f81f452bf3","role":"sequence","name":"Another version of Fibonacci-Pascal triangle A037027.","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:   1,     1;\n  2:   3,     7,       2;\n  3:   5,    27,      20,       3;\n  4:  11,   113,     156,      55,       5;\n  5:  21,   419,     988,     623,     130,       8;\n  6:  43,  1581,    5944,    5941,    2129,     293,     13;\n  7:  85,  5803,   33804,   50491,   27704,    6406,    626,    21;\n  8: 171, 21349,  188072,  403721,  318235,  109935,  17747,  1295,   34;\n  9: 341, 78027, 1026116, 3089011, 3364730, 1629836, 386608, 46041, 2605, 55;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+4*x+x^2)*P(n-2,x) + (4*x+2*x^2)*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) + (1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[(2 + 4*x + x^2)*P[n - 2, x] + (4*x + 2*x^2)*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}] + (1 + x)*P[n - 1, x] + 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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 4*x + x^2)*v[i - 2] + (4*x + 2*x^2)*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), x) + (1 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-e7c079f81f452bf3","steps":["basis A008277"],"edges":["e-0c8a8985960dccd5"],"text":"A111006 \u2192 basis A008277"},"headline":"A111006 \u2192 basis A008277","chain":null}