{"id":"n-81b00881f7f71483","revision":0,"first_index":0,"fingerprint":"f6-c1122197787ef9b0","name":"shift_x_plus_one of basis_map_A013611 of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["11","8"],["21","16"],["131","184","64"],["341","512","192"],["1651","3400","2304","512"],["5061","11248","8320","2048"],["21571","58456","58560","25600","4096"],["72181","211424","231744","112640","20480"],["287891","968552","1284992","837120","266240","32768"],["1009701","3660240","5293824","3817472","1372160","196608"],["3888611","15648888","25892160","22468608","10731520","2654208","262144"],["13985621","60328896","108112320","102993920","54992896","15597568","1835008"],["52871731","247926664","492225024","534817280","342056960","127991808","25690112","2097152"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","19","37","379","1045","7867","26677","168283","648469","3677563","15350005","81546139","357846229","1825676731"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["4"]]},"terms":[{"coeff":{"coeffs":[["-80","-64"],["84","64"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-205","-324","-128"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175620,"position":0,"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-6fce2e840d135f3d"]},{"id":175621,"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-6fce2e840d135f3d"]},{"id":175622,"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-6fce2e840d135f3d"]}],"region_exact":{},"incoming_edges":[{"id":"e-6fce2e840d135f3d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-81b00881f7f71483","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":"0e8edc99e3b046c027a11f7f971b5968416a01402439983b6255513a07d1187d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-af7a69a897cce4e9","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","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-5562eb8f7260f804","kind":"candidate_equal","node_a_id":"n-81b00881f7f71483","node_b_id":"n-b74340a4f1cd3ad2","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-b74340a4f1cd3ad2","other":{"id":"n-b74340a4f1cd3ad2","name":"basis_map_A007318 of basis_map_A013611 of Triangle read by rows: T(n,k) is the number of ways to tile a 3 X n rectangle with k pieces of 2 X 2 tiles and 3n-4k pieces of 1 X 1 tiles (0 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-c1122197787ef9b0","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["11","8"]],"card_rows":[["1"],["1"],["11","8"],["21","16"]],"row_sums_preview":["1","1","19","37","379","1045","7867","26677"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e3d901814302c059","steps":["basis A013611","basis A007318"],"edges":["e-099e290d18ba9894","e-69637a0f48d7969f"],"text":"A128099 \u2192 basis A013611 \u2192 basis A007318"},"headline":"A128099 \u2192 basis A013611 \u2192 basis A007318","chain":null}}],"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:     1;\n  1:     1;\n  2:    11,      8;\n  3:    21,     16;\n  4:   131,    184,     64;\n  5:   341,    512,    192;\n  6:  1651,   3400,   2304,    512;\n  7:  5061,  11248,   8320,   2048;\n  8: 21571,  58456,  58560,  25600,  4096;\n  9: 72181, 211424, 231744, 112640, 20480;\n","recurrence_sentence":"The row polynomials satisfy 4*n*P(n,x) = (84*n-80+(64*n-64)*x)*P(n-1,x) - (205+324*x+128*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((84*n - 80 + (64*n - 64)*x)*P[n - 1, x] - (205 + 324*x + 128*x^2)*D[P[n - 1, x], x])/(4*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((84*n - 80 + (64*n - 64)*x)*v[i - 1] - (205 + 324*x + 128*x^2)*deriv(v[i - 1], x))/(4*n)); 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-e3d901814302c059","steps":["basis A013611","x \u21a6 x+1"],"edges":["e-099e290d18ba9894","e-6fce2e840d135f3d"],"text":"A128099 \u2192 basis A013611 \u2192 x \u21a6 x+1"},"headline":"A128099 \u2192 basis A013611 \u2192 x \u21a6 x+1","chain":null}