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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"fd3f79a9655f7ab8f118fecdae52c11627efb962a5e881050e2f3a7cdef84999","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-9b5c7e97e5191da6","role":"sequence","name":"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:     4,      1;\n  2:    14,      9,       1;\n  3:    48,     57,      15,       1;\n  4:   164,    315,     145,      22,      1;\n  5:   560,   1625,    1155,     297,     30,      1;\n  6:  1912,   8055,    8265,    3190,    535,     39,     1;\n  7:  6528,  38937,   55335,   30001,   7410,    884,    49,    1;\n  8: 22288, 185103,  354417,  258962,  88211,  15288,  1372,   60,  1;\n  9: 76096, 869929, 2200935, 2107149, 949830, 224035, 28910, 2030, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -2*P(n-2,x) + (4+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[-2*P[n - 2, x] + (4 + 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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -2*v[i - 2] + (4 + 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-9b5c7e97e5191da6","steps":["basis A008277"],"edges":["e-7622699fc8055eaa"],"text":"A110441 \u2192 basis A008277"},"headline":"A110441 \u2192 basis A008277","chain":null}