{"id":"n-07c6a4edd51718c6","revision":1,"first_index":0,"fingerprint":"f6-4309d584dc07719c","name":"basis_map_A013611 of Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["11","56","64"],["41","304","704","512"],["153","1488","5184","7680","4096"],["571","6872","31936","71680","77824","32768"],["2131","30568","177536","534528","880640","753664","262144"],["7953","132448","922752","3486720","7720960","10027008","7077888","2097152"],["29681","562848","4573056","20794368","57896960","101122048","108265472","65011712","16777216"],["110771","2356392","21872256","116275200","390221824","857636864","1234960384","1124073472","587202560","134217728"],["413403","9748888","101767104","619284480","2433191936","6451200000","11692670976","14310965248","11324620800","5234491392","1073741824"],["1542841","39946384","463187264","3174999552","14296821760","44412698624","97145323520","149661155328","159199002624","111400714240","46170898432","8589934592"],["5757961","162379376","2070553024","15786211840","80154091520","285574168576","732190212096","1361496244224","1822760632320","1713960386560","1074815565824","403726925824","68719476736"],["21489003","655634808","9118059840","76534272000","432609239040","1739116707840","5116208873472","11153845518336","18023813480448","21326525890560","17964774457344","10204842295296","3504693313536","549755813888"],["80198051","2632071880","39646764800","363295354880","2262557040640","10131766575104","33645578944512","84183556816896","159503257436160","227782651019264","241396489388032","184133837914112","95588792139776","30236569763840","4398046511104"],["299303201","10514237120","170525574400","1693821265920","11523981762560","56888405917696","210520239505408","594745013305344","1293457670799360","2165830137675776","2769882391248896","2657502424465408","1851921178558464","885106860359680","259484744155136","35184372088832"],["1117014753","41819302208","726569429760","7776194304000","57383940136960","309613711196160","1263542626418688","3978958412447744","9772287532204032","18783199266078720","28164774177013760","32604934989938688","28572115317817344","18325560300142592","8114395812986880","2216615441596416","281474976710656"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["131","1561","18601","221651","2641211","31472881","375033361","4468927451","53252096051","634556225161","7561422605881","90102515045411","1073668757939051","12793922580223201","152453402204739361"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["11","56","64"],["41","304","704","512"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168040,"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-ab96494d3bfaa8ca"]},{"id":168041,"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-ab96494d3bfaa8ca"]},{"id":168042,"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-ab96494d3bfaa8ca"]},{"id":168043,"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-ab96494d3bfaa8ca"]}],"region_exact":{},"incoming_edges":[{"id":"e-ab96494d3bfaa8ca","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-07c6a4edd51718c6","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":"77f325e9d436cfed02c4c141259ca3894d3630d6ce996ee341bdc00c5dd6aed7","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-8eb9bb3a09f0b7fd","role":"sequence","name":"Triangle read by rows: T(n,k) number of tilings of a 2n X 3 grid by dominoes, 2k of which are in a vertical position (0<=k<=n).","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-1c3a35f297ef05fa","kind":"oeis_match","node_a_id":"n-07c6a4edd51718c6","node_b_id":null,"fingerprint":null,"oeis_id":"A077417","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":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A077417","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A077417","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:      11,       56,        64;\n  1:      41,      304,       704,        512;\n  2:     153,     1488,      5184,       7680,        4096;\n  3:     571,     6872,     31936,      71680,       77824,       32768;\n  4:    2131,    30568,    177536,     534528,      880640,      753664,      262144;\n  5:    7953,   132448,    922752,    3486720,     7720960,    10027008,     7077888,      2097152;\n  6:   29681,   562848,   4573056,   20794368,    57896960,   101122048,   108265472,     65011712,     16777216;\n  7:  110771,  2356392,  21872256,  116275200,   390221824,   857636864,  1234960384,   1124073472,    587202560,    134217728;\n  8:  413403,  9748888, 101767104,  619284480,  2433191936,  6451200000, 11692670976,  14310965248,  11324620800,   5234491392,  1073741824;\n  9: 1542841, 39946384, 463187264, 3174999552, 14296821760, 44412698624, 97145323520, 149661155328, 159199002624, 111400714240, 46170898432, 8589934592;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 11 + 56*x + 64*x^2, P(1,x) = 41 + 304*x + 704*x^2 + 512*x^3.","mathematica":"P[0, x_] := 11 + 56*x + 64*x^2;\nP[1, x_] := 41 + 304*x + 704*x^2 + 512*x^3;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + 8*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] = 11 + 56*x + 64*x^2; v[2] = 41 + 304*x + 704*x^2 + 512*x^3; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + 8*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-8eb9bb3a09f0b7fd","steps":["basis A013611"],"edges":["e-ab96494d3bfaa8ca"],"text":"A123519 \u2192 basis A013611"},"headline":"A123519 \u2192 basis A013611","chain":null}