{"id":"n-1e01398e35510f54","revision":0,"first_index":1,"fingerprint":"f6-d44a70ca53ce7ac3","name":"basis_map_A013609 of hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["8","16","4"],["32","96","72","8"],["128","512","640","256","16"],["512","2560","4480","3200","800","32"],["2048","12288","27648","28672","13440","2304","64"],["8192","57344","157696","215040","150528","50176","6272","128"],["32768","262144","851968","1441792","1351680","688128","172032","16384","256"],["131072","1179648","4423680","8945664","10543104","7299072","2838528","552960","41472","512"],["524288","5242880","22282240","52428800","74547200","65601536","35143680","10813440","1689600","102400","1024"],["2097152","23068672","109576192","294125568","490209280","524812288","360808448","154632192","38658048","4956160","247808","2048"],["8388608","100663296","528482304","1593835520","3048210432","3850371072","3244294144","1799356416","632586240","131203072","14057472","589824","4096"],["33554432","436207616","2508193792","8396996608","18129879040","26417823744","26417823744","18075353088","8284536832","2436628480","426409984","38764544","1384448","8192"],["134217728","1879048192","11744051200","43218108416","103993573376","171815469056","199149748224","162571223040","92462383104","35687235584","8921808896","1335885824","104366080","3211264","16384"],["536870912","8053063680","54358179840","218103808000","578813952000","1069648183296","1411341352960","1341212590080","914463129600","440297062400","145298030592","31284264960","4055367680","275251200","7372800","32768"],["2147483648","34359738368","249108103168","1082331758592","3140694835200","6420976107520","9507983851520","10322953895936","8226103885824","4768755875840","1972530839552","563580239872","105671294976","11978932224","713031680","16777216","65536"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["28","208","1552","11584","86464","645376","4817152","35955712","268377088","2003193856","14952042496","111603564544","833020346368","6217748512768","46409906716672"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["8","16","4"],["32","96","72","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199792,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-42bba86233fadd35"]},{"id":199793,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-42bba86233fadd35"]},{"id":199794,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-42bba86233fadd35"]},{"id":199795,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-42bba86233fadd35"]}],"region_exact":{},"incoming_edges":[{"id":"e-42bba86233fadd35","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-1e01398e35510f54","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":"1b3e9b7beb919c19cb9e30f7b6c1050c6bc88f25b1141a58166048be003c4499","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-dbc23e13bb290987","role":"sequence","name":"hadamard_factor parent of Triangle read by rows: t(n,m) = binomial(2*n,2*m) * binomial(n,m).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-06f23cac98cc8962","kind":"fingerprint_related","node_a_id":"n-180b0ec0ce696564","node_b_id":"n-1e01398e35510f54","fingerprint":"f6-d44a70ca53ce7ac3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d44a70ca53ce7ac3"},"other_id":"n-180b0ec0ce696564","other":{"id":"n-180b0ec0ce696564","name":"reversal of basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-d44a70ca53ce7ac3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["4","16","8"]],"card_rows":[["1"],["2","2"],["4","16","8"],["8","72","96","32"]],"row_sums_preview":["1","4","28","208","1552","11584","86464","645376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013609","reverse"],"edges":["e-55a1228129c4451e","e-f35753600c5398d3"],"text":"A086645 \u2192 basis A013609 \u2192 reverse"},"headline":"A086645 \u2192 basis A013609 \u2192 reverse","chain":null}},{"id":"l-1dcc883f3526a002","kind":"fingerprint_related","node_a_id":"n-91f141c4f7274be3","node_b_id":"n-1e01398e35510f54","fingerprint":"f6-d44a70ca53ce7ac3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d44a70ca53ce7ac3"},"other_id":"n-91f141c4f7274be3","other":{"id":"n-91f141c4f7274be3","name":"basis_map_A013609 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-d44a70ca53ce7ac3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["8","16","4"]],"card_rows":[["1"],["2","2"],["8","16","4"],["32","96","72","8"]],"row_sums_preview":["1","4","28","208","1552","11584","86464","645376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013609"],"edges":["e-55a1228129c4451e"],"text":"A086645 \u2192 basis A013609"},"headline":"A086645 \u2192 basis A013609","chain":null}},{"id":"l-730bb832f3248cb4","kind":"fingerprint_related","node_a_id":"n-cdf01fb0e5c8c5bd","node_b_id":"n-1e01398e35510f54","fingerprint":"f6-d44a70ca53ce7ac3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d44a70ca53ce7ac3"},"other_id":"n-cdf01fb0e5c8c5bd","other":{"id":"n-cdf01fb0e5c8c5bd","name":"reversal of shift_x_plus_one of chebyshev_t","first_index":0,"fingerprint":"f6-d44a70ca53ce7ac3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["2","4","1"]],"card_rows":[["1"],["1","1"],["2","4","1"],["4","12","9","1"]],"row_sums_preview":["1","2","7","26","97","362","1351","5042"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9ae8d431f7b33f22","steps":["x \u21a6 x+1","reverse"],"edges":["e-298ee5f2391f30e9","e-3fb9b29dd9d36a7e"],"text":"chebyshev_t \u2192 x \u21a6 x+1 \u2192 reverse"},"headline":"chebyshev_t \u2192 x \u21a6 x+1 \u2192 reverse","chain":null}},{"id":"l-754f457e89fb07d6","kind":"fingerprint_related","node_a_id":"n-55515e5d92a2ea3a","node_b_id":"n-1e01398e35510f54","fingerprint":"f6-d44a70ca53ce7ac3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d44a70ca53ce7ac3"},"other_id":"n-55515e5d92a2ea3a","other":{"id":"n-55515e5d92a2ea3a","name":"basis_map_A013609 of veronese_section_2_0 of Even-numbered rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-d44a70ca53ce7ac3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","2"],["8","16","4"]],"card_rows":[["1"],["2","2"],["8","16","4"],["32","96","72","8"]],"row_sums_preview":["1","4","28","208","1552","11584","86464","645376"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-08f49539099949d5","steps":["Veronese 2","basis A013609"],"edges":["e-f72875fb5d524bf0","e-8c0c84171cd9be3c"],"text":"A034870 \u2192 Veronese 2 \u2192 basis A013609"},"headline":"A034870 \u2192 Veronese 2 \u2192 basis A013609","chain":null}},{"id":"l-d864962fc4beec0e","kind":"fingerprint_related","node_a_id":"n-5fc5274bb6446a33","node_b_id":"n-1e01398e35510f54","fingerprint":"f6-d44a70ca53ce7ac3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-d44a70ca53ce7ac3"},"other_id":"n-5fc5274bb6446a33","other":{"id":"n-5fc5274bb6446a33","name":"shift_x_plus_one of chebyshev_t","first_index":0,"fingerprint":"f6-d44a70ca53ce7ac3","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","1"],["1","4","2"]],"card_rows":[["1"],["1","1"],["1","4","2"],["1","9","12","4"]],"row_sums_preview":["1","2","7","26","97","362","1351","5042"],"regions":["(-2, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9ae8d431f7b33f22","steps":["x \u21a6 x+1"],"edges":["e-298ee5f2391f30e9"],"text":"chebyshev_t \u2192 x \u21a6 x+1"},"headline":"chebyshev_t \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       8,       16,         4;\n   2:      32,       96,        72,         8;\n   3:     128,      512,       640,       256,        16;\n   4:     512,     2560,      4480,      3200,       800,        32;\n   5:    2048,    12288,     27648,     28672,     13440,      2304,        64;\n   6:    8192,    57344,    157696,    215040,    150528,     50176,      6272,       128;\n   7:   32768,   262144,    851968,   1441792,   1351680,    688128,    172032,     16384,      256;\n   8:  131072,  1179648,   4423680,   8945664,  10543104,   7299072,   2838528,    552960,    41472,     512;\n   9:  524288,  5242880,  22282240,  52428800,  74547200,  65601536,  35143680,  10813440,  1689600,  102400,   1024;\n  10: 2097152, 23068672, 109576192, 294125568, 490209280, 524812288, 360808448, 154632192, 38658048, 4956160, 247808, 2048;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -4*x^2*P(n-2,x) + (4+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 8 + 16*x + 4*x^2, P(2,x) = 32 + 96*x + 72*x^2 + 8*x^3.","mathematica":"P[1, x_] := 8 + 16*x + 4*x^2;\nP[2, x_] := 32 + 96*x + 72*x^2 + 8*x^3;\nP[n_, x_] := P[n, x] = Expand[-4*x^2*P[n - 2, x] + (4 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 8 + 16*x + 4*x^2; v[2] = 32 + 96*x + 72*x^2 + 8*x^3; for(n = 3, N, my(i = n); v[i] = -4*x^2*v[i - 2] + (4 + 4*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-dbc23e13bb290987","steps":["basis A013609"],"edges":["e-42bba86233fadd35"],"text":"Hadamard-factor parent of A155495 \u2192 basis A013609"},"headline":"Hadamard-factor parent of A155495 \u2192 basis A013609","chain":null}