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a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"1d831f4f36d8a47f427f8639eb94cb0e1fcfc11202d77831c28070c1666a8b2a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-87f3394ac59b064b","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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;\n  2:      5,       4;\n  3:     13,      12;\n  4:     73,     120,      48;\n  5:    281,     520,     240;\n  6:   1741,    4380,    3600,     960;\n  7:   8485,   23604,   21840,    6720;\n  8:  57233,  194992,  245280,  134400,  26880;\n  9: 328753, 1221840, 1699488, 1048320, 241920;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-4+(4*n-4)*x)*P(n-2,x) + P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4*n - 4 + (4*n - 4)*x)*P[n - 2, x] + P[n - 1, 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; for(n = 2, N, my(i = n + 1); v[i] = (4*n - 4 + (4*n - 4)*x)*v[i - 2] + v[i - 1]); 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-c914b69838ac70ec","steps":["basis A007318","basis A013609"],"edges":["e-6629899d205ea17b","e-c3a0267496cb0eee"],"text":"A059344 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A059344 \u2192 basis A007318 \u2192 basis A013609","chain":null}
