{"id":"n-6b39163848c2036c","revision":1,"first_index":0,"fingerprint":"f6-e309ee14657f718e","name":"basis_map_A013610 of Triangle read by rows, A007318 rows repeated.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["1"],["2","3"],["2","3"],["4","12","9"],["4","12","9"],["8","36","54","27"],["8","36","54","27"],["16","96","216","216","81"],["16","96","216","216","81"],["32","240","720","1080","810","243"],["32","240","720","1080","810","243"],["64","576","2160","4320","4860","2916","729"],["64","576","2160","4320","4860","2916","729"],["128","1344","6048","15120","22680","20412","10206","2187"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","5","5","25","25","125","125","625","625","3125","3125","15625","15625","78125"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166944,"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-c6857bd156c16708"]},{"id":166945,"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-c6857bd156c16708"]},{"id":166946,"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-c6857bd156c16708"]}],"region_exact":{},"incoming_edges":[{"id":"e-c6857bd156c16708","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-6b39163848c2036c","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"bf4f112c124f033a4bc631cd633d155443834688607b4cc92cff75a1ca04e08b","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-4124f4242e7d4c67","role":"sequence","name":"Triangle read by rows, A007318 rows repeated.","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-4b864d10de7a9f31","kind":"fingerprint_related","node_a_id":"n-6b39163848c2036c","node_b_id":"n-a9e3a6c6145e13ca","fingerprint":"f6-e309ee14657f718e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e309ee14657f718e"},"other_id":"n-a9e3a6c6145e13ca","other":{"id":"n-a9e3a6c6145e13ca","name":"basis_map_A013610 of Triangle read by rows: T(n,k) is the number of binary words of length n having degree of asymmetry equal to k (n >= 0; 0 <= k <= n/2).","first_index":0,"fingerprint":"f6-e309ee14657f718e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2"],["4","6"]],"card_rows":[["1"],["2"],["4","6"],["8","12"]],"row_sums_preview":["1","2","10","20","100","200","1000","2000"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b6aca576a689e9f0","steps":["basis A013610"],"edges":["e-4e3482570d7ce009"],"text":"A274496 \u2192 basis A013610"},"headline":"A274496 \u2192 basis A013610","chain":null}},{"id":"l-5c2aa6037194d963","kind":"fingerprint_related","node_a_id":"n-4e8690098ee9d7b4","node_b_id":"n-6b39163848c2036c","fingerprint":"f6-e309ee14657f718e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e309ee14657f718e"},"other_id":"n-4e8690098ee9d7b4","other":{"id":"n-4e8690098ee9d7b4","name":"shift_x_plus_one of Triangle T(n,k) read by rows: number of size-k antisymmetric subsets of {1,2,...,n}.","first_index":0,"fingerprint":"f6-e309ee14657f718e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["3","2"]],"card_rows":[["1"],["1"],["3","2"],["3","2"]],"row_sums_preview":["1","1","5","5","25","25","125","125"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ba3bb18490b15e2e","steps":["basis A007318"],"edges":["e-0b9b63070f7f0adc"],"text":"A188440 \u2192 basis A007318"},"headline":"A188440 \u2192 basis A007318","chain":null}},{"id":"l-b766d2348fa2301b","kind":"fingerprint_related","node_a_id":"n-1a86492085c1e21c","node_b_id":"n-6b39163848c2036c","fingerprint":"f6-e309ee14657f718e","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-e309ee14657f718e"},"other_id":"n-1a86492085c1e21c","other":{"id":"n-1a86492085c1e21c","name":"basis_map_A007318 of Triangle read by rows: T(n,k) is the number of ternary words of length n having degree of asymmetry equal to k (n>=0; 0<=k<=n/2).","first_index":0,"fingerprint":"f6-e309ee14657f718e","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3"],["9","6"]],"card_rows":[["1"],["3"],["9","6"],["27","18"]],"row_sums_preview":["1","3","15","45","225","675","3375","10125"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7c641d81e21514ce","steps":["x \u21a6 x+1"],"edges":["e-94bbecf72edd8f15"],"text":"A274498 \u2192 x \u21a6 x+1"},"headline":"A274498 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-039a74fb70cc66d1","kind":"oeis_match","node_a_id":"n-6b39163848c2036c","node_b_id":null,"fingerprint":null,"oeis_id":"A056451","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A056451","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-61b5b77114ebea5d","kind":"oeis_match","node_a_id":"n-6b39163848c2036c","node_b_id":null,"fingerprint":null,"oeis_id":"A074872","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A074872","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-a79c8eac0d579b3d","kind":"oeis_match","node_a_id":"n-6b39163848c2036c","node_b_id":null,"fingerprint":null,"oeis_id":"A162962","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A162962","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A056451","relation":"match_row_sums"},{"oeis_id":"A074872","relation":"match_row_sums"},{"oeis_id":"A162962","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-6b39163848c2036c","oeis_id":"A056451","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":33,"oeis_terms_available":33,"term_count":200,"raw_size":7981},{"node_id":"n-6b39163848c2036c","oeis_id":"A074872","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":33,"oeis_terms_available":33,"term_count":200,"raw_size":7914},{"node_id":"n-6b39163848c2036c","oeis_id":"A162962","target":"row_sums","oeis_offset":1,"first_row":1,"rows_used":200,"terms_matched":14,"oeis_terms_agreeing":30,"oeis_terms_available":30,"term_count":200,"raw_size":7983}],"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:  2,  3;\n  3:  2,  3;\n  4:  4, 12,   9;\n  5:  4, 12,   9;\n  6:  8, 36,  54,  27;\n  7:  8, 36,  54,  27;\n  8: 16, 96, 216, 216, 81;\n  9: 16, 96, 216, 216, 81;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+3*x)*P(n-2,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[(2 + 3*x)*P[n - 2, 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] = (2 + 3*x)*v[i - 2]); 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-4124f4242e7d4c67","steps":["basis A013610"],"edges":["e-c6857bd156c16708"],"text":"A152198 \u2192 basis A013610"},"headline":"A152198 \u2192 basis A013610","chain":null}
