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so every real-rooted input has a real-rooted image","lean_route":null,"lean_declarations":[],"certificate_hash":"b5e854c485fcb260c4273ee077af6ef706a6969fd16c76559b276b2dbea9b4ae","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","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":[{"id":"l-5a4807be1e9464b9","kind":"oeis_match","node_a_id":"n-95fc06ddb38bf951","node_b_id":null,"fingerprint":null,"oeis_id":"A060821","in_catalog":1,"target":"triangle","row_alignment":"tabl","rows_matched":11,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":11,"target":"triangle","terms_compared":66},"in_catalog":true,"oeis_id":"A060821","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-95550ad3de05f258","kind":"oeis_match","node_a_id":"n-95fc06ddb38bf951","node_b_id":null,"fingerprint":null,"oeis_id":"A062267","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A062267","oeis_nonn":false},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A060821","relation":"identification"},{"oeis_id":"A062267","relation":"match_row_sums"},{"oeis_id":"A060821","relation":"match_triangle"}],"bfiles":[{"node_id":"n-95fc06ddb38bf951","oeis_id":"A060821","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":66,"oeis_terms_agreeing":66,"oeis_terms_available":66,"term_count":990,"raw_size":14808},{"node_id":"n-95fc06ddb38bf951","oeis_id":"A062267","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":24,"oeis_terms_available":24,"term_count":200,"raw_size":20509}],"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:    0,     2;\n  2:   -2,     0,      4;\n  3:    0,   -12,      0,      8;\n  4:   12,     0,    -48,      0,    16;\n  5:    0,   120,      0,   -160,     0,    32;\n  6: -120,     0,    720,      0,  -480,     0,    64;\n  7:    0, -1680,      0,   3360,     0, -1344,     0,   128;\n  8: 1680,     0, -13440,      0, 13440,     0, -3584,     0, 256;\n  9:    0, 30240,      0, -80640,     0, 48384,     0, -9216,   0, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, 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; for(n = 1, N, my(i = n + 1); v[i] = 2*x*v[i - 1] - 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}},{"oeis_id":"A060821","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:    1;\n  1:    0,     2;\n  2:   -2,     0,      4;\n  3:    0,   -12,      0,      8;\n  4:   12,     0,    -48,      0,    16;\n  5:    0,   120,      0,   -160,     0,    32;\n  6: -120,     0,    720,      0,  -480,     0,    64;\n  7:    0, -1680,      0,   3360,     0, -1344,     0,   128;\n  8: 1680,     0, -13440,      0, 13440,     0, -3584,     0, 256;\n  9:    0, 30240,      0, -80640,     0, 48384,     0, -9216,   0, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*x*P(n-1,x) - d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[2*x*P[n - 1, x] - D[P[n - 1, x], x]];\nTable[PadRight[CoefficientList[P[n, x], x], n + 1], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = 2*x*v[i - 1] - deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); 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":true}}],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["basis A060821"],"edges":["e-5f70bba9f74a761e"],"text":"A023531 \u2192 basis A060821"},"headline":"A060821: Triangle read by rows. 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