{"id":"n-7967b266138e041c","revision":0,"first_index":0,"fingerprint":"f6-f5bc6b0dc876ed19","name":"shift_x_plus_one of basis_map_A007318 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":[["29","22","4"],["169","188","68","8"],["985","1444","780","184","16"],["5741","10446","7500","2656","464","32"],["33461","72714","65112","30752","8080","1120","64"],["195025","492760","528600","312080","109520","22848","2624","128"],["1136689","3273896","4092008","2898928","1273200","355008","61376","6016","256"],["6625109","21423994","30571240","25265504","13327536","4653600","1075648","158720","13568","512"],["38613965","138520286","222183420","209836576","129223024","54221664","15699712","3097600","398592","30208","1024"],["225058681","886925652","1579569852","1678120792","1181683760","579122432","201565952","49826304","8573184","977920","66560","2048"],["1311738121","5633150988","11029086996","13018027880","10317121120","5783880448","2351940864","698992128","150693120","22983680","2354176","145408","4096"],["7645370045","35535456518","75861254100","98488220480","86757098720","54758402496","25477840128","8848008192","2293569792","438310400","60025856","5578752","315392","8192"],["44560482149","222870328210","515209350640","729633803200","707201912160","496280731968","260031904896","103344737280","31306742016","7194018304","1234421760","153378816","13045760","679936","16384"],["259717522849","1390807477040","3461135506160","5309733300000","5615721980640","4337329813632","2527275053184","1131284225280","392236356864","105339283456","21734541312","3383537664","384716800","30162944","1458176","32768"],["1513744654945","8641409579728","23033218640400","38051037009120","43606596571680","36759142111104","23578278041472","11738910720768","4584679849728","1409314396160","339851393024","63616929792","9062330368","949731328","69058560","3112960","65536"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["55","433","3409","26839","211303","1663585","13097377","103115431","811826071","6391493137","50320119025","396169459063","3119035553479","24556114968769","193329884196673"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["29","22","4"]],"recurrence":{"denominator":{"coeffs":[["3"],["2"]]},"terms":[{"coeff":{"coeffs":[["11","4"],["6","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["16","12","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},{"position":1,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":"e-8d5faac942c17aea","recurrence":{"first_index":0,"initial_polynomials":[["29","22","4"],["169","188","68","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184727,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-4, -2)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":false,"value":"-2"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-4, -2)","region_exact":null,"edges":["e-795f3a062a4184d5"]},{"id":184728,"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-764191a5e4d1c155","e-795f3a062a4184d5"]},{"id":184729,"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-764191a5e4d1c155","e-795f3a062a4184d5"]},{"id":184730,"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-764191a5e4d1c155","e-795f3a062a4184d5"]},{"id":184731,"position":4,"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-764191a5e4d1c155"]}],"region_exact":{},"incoming_edges":[{"id":"e-764191a5e4d1c155","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-7967b266138e041c","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":"fc4fd40c06d2d3bc5cd4ce3f31164ebf967cf09694b335bea010f82fd7d55e92","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-47b361d1e904fbeb","role":"sequence","name":"basis_map_A007318 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).","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-8d5faac942c17aea","operation":{"operation":"definition_equality"},"operation_detail":null,"conclusion_id":"n-7967b266138e041c","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":"2d4b9ebed6a3efd473d4898e88487ac53eeb5ac75beeea9627b86b7128c9987b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[],"premises":[]},{"id":"e-795f3a062a4184d5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-7967b266138e041c","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":"14514ab44fd2ce0aa17350ef1a8233618e4e9249e35d9311d151a216113482af","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-4"},"upper":{"closed":false,"value":"-2"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-47b361d1e904fbeb","role":"sequence","name":"basis_map_A007318 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).","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":false,"value":"-3"},"upper":{"closed":false,"value":"-1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-3, -1)"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"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:        29,        22,          4;\n  1:       169,       188,         68,          8;\n  2:       985,      1444,        780,        184,         16;\n  3:      5741,     10446,       7500,       2656,        464,        32;\n  4:     33461,     72714,      65112,      30752,       8080,      1120,        64;\n  5:    195025,    492760,     528600,     312080,     109520,     22848,      2624,      128;\n  6:   1136689,   3273896,    4092008,    2898928,    1273200,    355008,     61376,     6016,     256;\n  7:   6625109,  21423994,   30571240,   25265504,   13327536,   4653600,   1075648,   158720,   13568,    512;\n  8:  38613965, 138520286,  222183420,  209836576,  129223024,  54221664,  15699712,  3097600,  398592,  30208,  1024;\n  9: 225058681, 886925652, 1579569852, 1678120792, 1181683760, 579122432, 201565952, 49826304, 8573184, 977920, 66560, 2048;\n","recurrence_sentence":"The row polynomials satisfy (2*n+3)*P(n,x) = (6*n+11+(2*n+4)*x)*P(n-1,x) + (16+12*x+2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 29 + 22*x + 4*x^2.","mathematica":"P[0, x_] := 29 + 22*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((6*n + 11 + (2*n + 4)*x)*P[n - 1, x] + (16 + 12*x + 2*x^2)*D[P[n - 1, x], x])/(2*n + 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 29 + 22*x + 4*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((6*n + 11 + (2*n + 4)*x)*v[i - 1] + (16 + 12*x + 2*x^2)*deriv(v[i - 1], x))/(2*n + 3)); 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 A007318","basis A007318"],"edges":["e-1acefd6886502727","e-764191a5e4d1c155"],"text":"A123519 \u2192 basis A007318 \u2192 basis A007318"},"headline":"A123519 \u2192 basis A007318 \u2192 basis A007318","chain":null}