{"id":"n-84c761192e0db3c0","revision":1,"first_index":0,"fingerprint":"f6-23ef07238c102b65","name":"basis_map_A013611 of Riordan array (1-x,x/(1-x)).","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":true,"rows":[["1","8","16"],["2","20","64","64"],["4","48","208","384","256"],["8","112","608","1600","2048","1024"],["16","256","1664","5632","10496","10240","4096"],["32","576","4352","17920","43520","62464","49152","16384"],["64","1280","11008","53248","158720","299008","348160","229376","65536"],["128","2816","27136","150528","530432","1232896","1892352","1851392","1048576","262144"],["256","6144","65536","409600","1662976","4587520","8716288","11272192","9502720","4718592","1048576"],["512","13312","155648","1081344","4964352","15826944","35782656","57409536","64094208","47448064","20971520","4194304"],["1024","28672","364544","2785280","14254080","51511296","134873088","257949696","357826560","351272960","231735296","92274688","16777216"],["2048","61440","843776","7028736","39649280","160038912","475791360","1055391744","1747451904","2133852160","1868562432","1111490560","402653184","67108864"],["4096","131072","1933312","17432576","107413504","478674944","1591738368","4013948928","7716470784","11257511936","12272533504","9697230848","5251268608","1744830464","268435456"],["8192","278528","4390912","42598400","284557312","1387003904","5098176512","14394851328","31488737280","53380907008","69575114752","68484595712","49291460608","24494735360","7516192768","1073741824"],["16384","589824","9895936","102760448","739508224","3912237056","15744368640","49182408704","120556879872","232716763136","352673857536","415269650432","372521304064","246155313152","113011326976","32212254720","4294967296"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["25","150","900","5400","32400","194400","1166400","6998400","41990400","251942400","1511654400","9069926400","54419558400","326517350400","1959104102400"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1","8","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167684,"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-d3bf3843e93b142d"]},{"id":167685,"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-d3bf3843e93b142d"]},{"id":167686,"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-d3bf3843e93b142d"]}],"region_exact":{},"incoming_edges":[{"id":"e-d3bf3843e93b142d","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-84c761192e0db3c0","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":"4e2e1b9582457221a5d9760d051bd428a769d955e6375814f9e51119566c60e5","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7717563017cb9e5f","role":"sequence","name":"Riordan array (1-x,x/(1-x)).","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-fd28443b5515afe5","kind":"oeis_match","node_a_id":"n-84c761192e0db3c0","node_b_id":null,"fingerprint":null,"oeis_id":"A055846","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":"A055846","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A055846","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:   1,     8,     16;\n  1:   2,    20,     64,      64;\n  2:   4,    48,    208,     384,     256;\n  3:   8,   112,    608,    1600,    2048,     1024;\n  4:  16,   256,   1664,    5632,   10496,    10240,     4096;\n  5:  32,   576,   4352,   17920,   43520,    62464,    49152,    16384;\n  6:  64,  1280,  11008,   53248,  158720,   299008,   348160,   229376,    65536;\n  7: 128,  2816,  27136,  150528,  530432,  1232896,  1892352,  1851392,  1048576,   262144;\n  8: 256,  6144,  65536,  409600, 1662976,  4587520,  8716288, 11272192,  9502720,  4718592,  1048576;\n  9: 512, 13312, 155648, 1081344, 4964352, 15826944, 35782656, 57409536, 64094208, 47448064, 20971520, 4194304;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 8*x + 16*x^2.","mathematica":"P[0, x_] := 1 + 8*x + 16*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + 4*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 + 8*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (2 + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7717563017cb9e5f","steps":["basis A013611"],"edges":["e-d3bf3843e93b142d"],"text":"A159854 \u2192 basis A013611"},"headline":"A159854 \u2192 basis A013611","chain":null}