{"id":"n-3d55772447340781","revision":2,"first_index":0,"fingerprint":"f6-9d7293d396b320fd","name":"basis_map_A007318 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle 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":false,"rows":[["1"],["4","2"],["16","16","3"],["64","96","40","4"],["256","512","336","80","5"],["1024","2560","2304","896","140","6"],["4096","12288","14080","7680","2016","224","7"],["16384","57344","79872","56320","21120","4032","336","8"],["65536","262144","430080","372736","183040","50688","7392","480","9"],["262144","1179648","2228224","2293760","1397760","512512","109824","12672","660","10"],["1048576","5242880","11206656","13369344","9748480","4472832","1281280","219648","20592","880","11"],["4194304","23068672","55050240","74711040","63504384","35094528","12673024","2928640","411840","32032","1144","12"],["16777216","100663296","265289728","403701760","392232960","254017536","111132672","32587776","6223360","732160","48048","1456","13"],["67108864","436207616","1258291200","2122317824","2321285120","1725825024","889061376","317521920","77395968","12446720","1244672","69888","1820","14"],["268435456","1879048192","5888802816","10905190400","13264486400","11142168576","6615662592","2794192896","833495040","171991040","23648768","2036736","99008","2240","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","35","204","1189","6930","40391","235416","1372105","7997214","46611179","271669860","1583407981","9228778026","53789260175"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":"e-507237b5f4039072","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-1"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161725,"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-24d6e61d1fff2e02"]},{"id":161726,"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-24d6e61d1fff2e02","e-c620f77b38a31b20"]},{"id":161727,"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-24d6e61d1fff2e02","e-c620f77b38a31b20"]},{"id":161728,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-24d6e61d1fff2e02","e-c620f77b38a31b20"]},{"id":161729,"position":4,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-c620f77b38a31b20"]}],"region_exact":{},"incoming_edges":[{"id":"e-24d6e61d1fff2e02","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-3d55772447340781","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":"e0ce1eefbe5d60c78ec1d07c946879edc9ec6ad4aa509e332b4dcf6ba70399bc","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-2245b35ec7cd4857","role":"sequence","name":"One half of odd-numbered entries of even-numbered rows of Pascal's triangle 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]"}]},{"id":"e-507237b5f4039072","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-3d55772447340781","premise_count":0,"theorem":"all-index identity Q_n = P_n certified by exact Ore-algebra division and identifying rows (polytool preserver-image certificate of the identity map)","lean_route":null,"lean_declarations":[],"certificate_hash":"cc04fe0bd47998e2282279cdce592cb5ec166bea7571f354de48fab40da9f32a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-c620f77b38a31b20","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3d55772447340781","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":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"3b9a59e73c5c6b171b34b66269d76d5c2c355677db43331bffc680882871a217","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2245b35ec7cd4857","role":"sequence","name":"One half of odd-numbered entries of even-numbered rows of Pascal's triangle 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-4768fc4906f9dec2","kind":"fingerprint_related","node_a_id":"n-3d55772447340781","node_b_id":"n-95c84497c8f9b78b","fingerprint":"f6-9d7293d396b320fd","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9d7293d396b320fd"},"other_id":"n-95c84497c8f9b78b","other":{"id":"n-95c84497c8f9b78b","name":"mobius_x_over_x_plus_one of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","first_index":0,"fingerprint":"f6-9d7293d396b320fd","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["3","16","16"]],"card_rows":[["1"],["2","4"],["3","16","16"],["4","40","96","64"]],"row_sums_preview":["1","6","35","204","1189","6930","40391","235416"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2245b35ec7cd4857","steps":["x \u21a6 x/(x+1)"],"edges":["e-d6e48ffab8c17735"],"text":"A091044 \u2192 x \u21a6 x/(x+1)"},"headline":"A091044 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-7b3755df092d5b18","kind":"oeis_match","node_a_id":"n-3d55772447340781","node_b_id":null,"fingerprint":null,"oeis_id":"A001109","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":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A001109","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A001109","relation":"match_row_sums"}],"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:      4,       2;\n  2:     16,      16,       3;\n  3:     64,      96,      40,       4;\n  4:    256,     512,     336,      80,       5;\n  5:   1024,    2560,    2304,     896,     140,      6;\n  6:   4096,   12288,   14080,    7680,    2016,    224,      7;\n  7:  16384,   57344,   79872,   56320,   21120,   4032,    336,     8;\n  8:  65536,  262144,  430080,  372736,  183040,  50688,   7392,   480,   9;\n  9: 262144, 1179648, 2228224, 2293760, 1397760, 512512, 109824, 12672, 660, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) + (4+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] + (4 + 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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] + (4 + 2*x)*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-2245b35ec7cd4857","steps":["basis A007318"],"edges":["e-24d6e61d1fff2e02"],"text":"A091044 \u2192 basis A007318"},"headline":"A091044 \u2192 basis A007318","chain":null}
