{"id":"n-964d853754e5a7e8","revision":1,"first_index":0,"fingerprint":"f6-298f80c143fc4de2","name":"reversal of A133080 * A007318.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["1","2"],["1","2","1"],["1","4","5","2"],["1","4","6","4","1"],["1","6","14","16","9","2"],["1","6","15","20","15","6","1"],["1","8","27","50","55","36","13","2"],["1","8","28","56","70","56","28","8","1"],["1","10","44","112","182","196","140","64","17","2"],["1","10","45","120","210","252","210","120","45","10","1"],["1","12","65","210","450","672","714","540","285","100","21","2"],["1","12","66","220","495","792","924","792","495","220","66","12","1"],["1","14","90","352","935","1782","2508","2640","2079","1210","506","144","25","2"],["1","14","91","364","1001","2002","3003","3432","3003","2002","1001","364","91","14","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","4","12","16","48","64","192","256","768","1024","3072","4096","12288","16384"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0","-3"],["0","2","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-5","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161161,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-7d9675c994dd8650"]},{"id":161162,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-7d9675c994dd8650"]},{"id":161163,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-7d9675c994dd8650"]}],"region_exact":{},"incoming_edges":[{"id":"e-7d9675c994dd8650","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-964d853754e5a7e8","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"cc7f203e5011d7933ee72281498ac7e401ff430470db005e71e2d5aae18149d8","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-d75669a8cd7f9393","role":"sequence","name":"A133080 * A007318.","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-ecd81e7f7a3f01aa","kind":"fingerprint_related","node_a_id":"n-d75669a8cd7f9393","node_b_id":"n-964d853754e5a7e8","fingerprint":"f6-298f80c143fc4de2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-298f80c143fc4de2"},"other_id":"n-d75669a8cd7f9393","other":{"id":"n-d75669a8cd7f9393","name":"A133080 * A007318.","first_index":0,"fingerprint":"f6-298f80c143fc4de2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","1"],["1","2","1"]],"card_rows":[["1"],["2","1"],["1","2","1"],["2","5","4","1"]],"row_sums_preview":["1","3","4","12","16","48","64","192"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A133087"],"derivation":{"root":"n-d75669a8cd7f9393","steps":[],"edges":[],"text":"A133087"},"headline":"A133087: A133080 * A007318.","chain":null}},{"id":"l-572fcd4272f8817a","kind":"oeis_match","node_a_id":"n-964d853754e5a7e8","node_b_id":null,"fingerprint":null,"oeis_id":"A133087","in_catalog":1,"target":"reversed_triangle","row_alignment":"tabl","rows_matched":10,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"tabl","rows_matched":10,"target":"reversed_triangle","terms_compared":55},"in_catalog":true,"oeis_id":"A133087","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-58fea4cd8041468a","kind":"oeis_match","node_a_id":"n-964d853754e5a7e8","node_b_id":null,"fingerprint":null,"oeis_id":"A084221","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"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":"A084221","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A133087","relation":"match_reversed_triangle"},{"oeis_id":"A084221","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-964d853754e5a7e8","oeis_id":"A084221","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":34,"oeis_terms_available":34,"term_count":200,"raw_size":6999},{"node_id":"n-964d853754e5a7e8","oeis_id":"A133087","target":"reversed_triangle","oeis_offset":0,"first_row":0,"rows_used":44,"terms_matched":55,"oeis_terms_agreeing":55,"oeis_terms_available":55,"term_count":990,"raw_size":10742}],"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,  2;\n  2: 1,  2,  1;\n  3: 1,  4,  5,   2;\n  4: 1,  4,  6,   4,   1;\n  5: 1,  6, 14,  16,   9,   2;\n  6: 1,  6, 15,  20,  15,   6,   1;\n  7: 1,  8, 27,  50,  55,  36,  13,  2;\n  8: 1,  8, 28,  56,  70,  56,  28,  8,  1;\n  9: 1, 10, 44, 112, 182, 196, 140, 64, 17, 2;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+2*n*x+(3*n-3)*x^2)*P(n-1,x) - (2*x+5*x^2+3*x^3)*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[(1 + 2*n*x + (3*n - 3)*x^2)*P[n - 1, x] - (2*x + 5*x^2 + 3*x^3)*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] = (1 + 2*n*x + (3*n - 3)*x^2)*v[i - 1] - (2*x + 5*x^2 + 3*x^3)*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":"A133087","target":"reversed_triangle","first_row":0,"oeis_offset":0,"reversed":true,"trim_zeros":"none","regular_triangle":true,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 2,  1;\n  2: 1,  2,  1;\n  3: 2,  5,  4,   1;\n  4: 1,  4,  6,   4,   1;\n  5: 2,  9, 16,  14,   6,   1;\n  6: 1,  6, 15,  20,  15,   6,   1;\n  7: 2, 13, 36,  55,  50,  27,   8,  1;\n  8: 1,  8, 28,  56,  70,  56,  28,  8,  1;\n  9: 2, 17, 64, 140, 196, 182, 112, 44, 10, 1;\n","recurrence_sentence":"Row n lists the coefficients of P(n,x) in descending powers of x, where P(n,x) = (1+2*n*x+(3*n-3)*x^2)*P(n-1,x) - (2*x+5*x^2+3*x^3)*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[(1 + 2*n*x + (3*n - 3)*x^2)*P[n - 1, x] - (2*x + 5*x^2 + 3*x^3)*D[P[n - 1, x], x]];\nTable[Reverse[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] = (1 + 2*n*x + (3*n - 3)*x^2)*v[i - 1] - (2*x + 5*x^2 + 3*x^3)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n + 1 - #c)); c = Vecrev(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":true,"trim_zeros":"none","regular_triangle":true}}],"derivation":{"root":"n-d75669a8cd7f9393","steps":["reverse"],"edges":["e-7d9675c994dd8650"],"text":"A133087 \u2192 reverse"},"headline":"A133087 \u2192 reverse","chain":null}