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so every real-rooted input has a real-rooted image","lean_route":null,"lean_declarations":[],"certificate_hash":"1e1492af5024be4114750c5d8ee039948d1ab798f3f45cc3f878537b36eac12a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-5fe240122e98ffb3","role":"sequence","name":"basis-map coefficients of A scaled Hermite triangle.","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":[{"id":"l-d0e5b48330c630a1","kind":"oeis_match","node_a_id":"n-0ce01d5a88917ba1","node_b_id":null,"fingerprint":null,"oeis_id":"A293604","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A293604","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A293604","relation":"match_row_sums"},{"oeis_id":"A112227","relation":"tail"}],"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:     -2,       0,     1;\n   2:      0,      -6,     0,      1;\n   3:     12,       0,   -12,      0,      1;\n   4:      0,      60,     0,    -20,      0,      1;\n   5:   -120,       0,   180,      0,    -30,      0,    1;\n   6:      0,    -840,     0,    420,      0,    -42,    0,    1;\n   7:   1680,       0, -3360,      0,    840,      0,  -56,    0,   1;\n   8:      0,   15120,     0, -10080,      0,   1512,    0,  -72,   0,    1;\n   9: -30240,       0, 75600,      0, -25200,      0, 2520,    0, -90,    0, 1;\n  10:      0, -332640,     0, 277200,      0, -55440,    0, 3960,   0, -110, 0, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x*P(n-1,x) - 2*d/dx P(n-1,x) for n >= 2, with P(1,x) = -2 + x^2.","mathematica":"P[1, x_] := -2 + x^2;\nP[n_, x_] := P[n, x] = Expand[x*P[n - 1, x] - 2*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] = -2 + x^2; for(n = 2, N, my(i = n); v[i] = x*v[i - 1] - 2*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-5fe240122e98ffb3","steps":["basis A112227"],"edges":["e-77b4dff64dd0f71c"],"text":"basis-map coefficients of A112227 \u2192 basis A112227"},"headline":"basis-map coefficients of A112227 \u2192 basis A112227","chain":null}