{"id":"n-0bad942bfffd4f03","revision":2,"first_index":0,"fingerprint":"f6-076e7110b92525fe","name":"basis_map_A007318 of Pascal-(1,4,1) array.","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"],["2","1"],["8","8","1"],["24","36","14","1"],["80","160","100","20","1"],["256","640","560","200","26","1"],["832","2496","2800","1440","336","32","1"],["2688","9408","12896","8720","3016","508","38","1"],["8704","34816","56384","47296","21856","5504","716","44","1"],["28160","126720","236800","237440","137952","46960","9120","960","50","1"],["91136","455680","965120","1126400","789952","341312","90080","14080","1240","56","1"],["294912","1622016","3840000","5114880","4207872","2212224","745792","158560","20600","1556","62","1"],["954368","5726208","14985216","22435840","21196032","13157376","5429376","1479552","261040","28896","1908","68","1"],["3088384","20074496","57544704","95676416","102118912","73191168","35848192","12005888","2718272","407456","39184","2296","74","1"],["9994240","69959680","218009600","398581760","474441728","385914880","219234560","87495680","24404800","4692928","609040","51680","2720","80","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","17","75","361","1683","7937","37275","175321","824163","3875057","18218475","85655881","402715443","1893393377"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","1"]]},"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-f5f82cd32c2e9500","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["11","2"],["-14","-1"],["5","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["50","53","4"],["-20","-20","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["20","40","21","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161605,"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-4bbb4fa4f4a05d5e"]},{"id":161606,"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-4bbb4fa4f4a05d5e","e-ac2b3ac5f11d35ca"]},{"id":161607,"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-4bbb4fa4f4a05d5e","e-ac2b3ac5f11d35ca"]},{"id":161608,"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-4bbb4fa4f4a05d5e","e-ac2b3ac5f11d35ca"]},{"id":161609,"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-ac2b3ac5f11d35ca"]}],"region_exact":{},"incoming_edges":[{"id":"e-4bbb4fa4f4a05d5e","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-0bad942bfffd4f03","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":"7347293f74c2a8b9063705e4015932c453652b5a4d5f8959d069e768320195f7","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-ed9434bfd889c409","role":"sequence","name":"Pascal-(1,4,1) array.","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-f5f82cd32c2e9500","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-0bad942bfffd4f03","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":"e81801225b8630e13135f4befc53a1610561219becaf9ca1b129d35482101ce7","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-ac2b3ac5f11d35ca","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-0bad942bfffd4f03","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":"3b04253781b094e2c57b7cb735f9de81622bbb699d3af5f95874f09fc55d4523","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-ed9434bfd889c409","role":"sequence","name":"Pascal-(1,4,1) array.","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-0a3270ef52419999","kind":"fingerprint_related","node_a_id":"n-0bad942bfffd4f03","node_b_id":"n-0821f66594bc073c","fingerprint":"f6-076e7110b92525fe","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-076e7110b92525fe"},"other_id":"n-0821f66594bc073c","other":{"id":"n-0821f66594bc073c","name":"mobius_x_over_x_plus_one of Pascal-(1,4,1) array.","first_index":0,"fingerprint":"f6-076e7110b92525fe","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","8","8"]],"card_rows":[["1"],["1","2"],["1","8","8"],["1","14","36","24"]],"row_sums_preview":["1","3","17","75","361","1683","7937","37275"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ed9434bfd889c409","steps":["x \u21a6 x/(x+1)"],"edges":["e-1bbdf11925fb622c"],"text":"A081579 \u2192 x \u21a6 x/(x+1)"},"headline":"A081579 \u2192 x \u21a6 x/(x+1)","chain":null}},{"id":"l-db4655fe630420bd","kind":"oeis_match","node_a_id":"n-0bad942bfffd4f03","node_b_id":null,"fingerprint":null,"oeis_id":"A015525","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":"A015525","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A015525","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:     2,      1;\n  2:     8,      8,      1;\n  3:    24,     36,     14,      1;\n  4:    80,    160,    100,     20,      1;\n  5:   256,    640,    560,    200,     26,     1;\n  6:   832,   2496,   2800,   1440,    336,    32,    1;\n  7:  2688,   9408,  12896,   8720,   3016,   508,   38,   1;\n  8:  8704,  34816,  56384,  47296,  21856,  5504,  716,  44,  1;\n  9: 28160, 126720, 236800, 237440, 137952, 46960, 9120, 960, 50, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+4*x)*P(n-2,x) + (2+x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(4 + 4*x)*P[n - 2, x] + (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] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 4*x)*v[i - 2] + (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-ed9434bfd889c409","steps":["basis A007318"],"edges":["e-4bbb4fa4f4a05d5e"],"text":"A081579 \u2192 basis A007318"},"headline":"A081579 \u2192 basis A007318","chain":null}
