{"id":"n-3bbccf23445ea940","revision":0,"first_index":0,"fingerprint":"f6-f01e0b1962cfad1f","name":"basis_map_A013611 of basis_map_A013611 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["6","16"],["41","208","256"],["281","2096","5120","4096"],["1926","18960","69120","110592","65536"],["13201","161440","782080","1875968","2228224","1048576"],["90481","1322336","7988480","25534464","45547520","42991616","16777216"],["620166","10542608","76294656","304680960","725155840","1028653056","805306368","268435456"],["4250681","82398576","694755840","3327946752","9905438720","18760073216","22078816256","14763950080","4294967296"],["29134601","634258320","6105373440","34107039744","121860063232","288778878976","453907578880","456340275200","266287972352","68719476736"],["199691526","4823563280","52190991360","333107306496","1388827639808","3952453091328","7775736299520","10442139238400","9161165242368","4741643894784","1099511627776"],["1368706081","36325749056","436408578560","3132699967488","14929650319360","49599634997248","117215495979008","197050415185920","230936096538624","179701431664640","83562883710976","17592186044416"],["9381251041","271355977408","3583881043456","28578329722880","153241924075520","582119396999168","1606326895509504","3244358702727168","4760198153502720","4948145922375680","3459063580983296","1460151441686528","281474976710656"],["64300051206","2013266109456","28992454364160","254257704787968","1515017093775360","6477106929205248","20440983124574208","48214690832056320","85000190221615104","110620490481008640","103300216941182976","65548485201494016","25332747903959040","4503599627370496"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","22","505","11593","266134","6109489","140252113","3219689110","73912597417","1696770051481","38951798586646","894194597441377","20527523942565025","471238856081554198"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":205390,"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-1c2a9398e8782140"]},{"id":205391,"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-1c2a9398e8782140"]},{"id":205392,"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-1c2a9398e8782140"]},{"id":205393,"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-1c2a9398e8782140"]}],"region_exact":{},"incoming_edges":[{"id":"e-1c2a9398e8782140","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-3bbccf23445ea940","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":"30f5080248663347df9a894dc662d4adbf177c056f117a05d6fa48d2d037e1ab","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a18927332224e18f","role":"sequence","name":"basis_map_A013611 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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":[],"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:       6,       16;\n  3:      41,      208,       256;\n  4:     281,     2096,      5120,       4096;\n  5:    1926,    18960,     69120,     110592,      65536;\n  6:   13201,   161440,    782080,    1875968,    2228224,     1048576;\n  7:   90481,  1322336,   7988480,   25534464,   45547520,    42991616,    16777216;\n  8:  620166, 10542608,  76294656,  304680960,  725155840,  1028653056,   805306368,   268435456;\n  9: 4250681, 82398576, 694755840, 3327946752, 9905438720, 18760073216, 22078816256, 14763950080, 4294967296;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+16*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 16*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + 16*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-0eba401b117f16a9","steps":["basis A013611","basis A013611"],"edges":["e-39137b152517cefd","e-1c2a9398e8782140"],"text":"A085478 \u2192 basis A013611 \u2192 basis A013611"},"headline":"A085478 \u2192 basis A013611 \u2192 basis A013611","chain":null}