{"id":"n-256337cf537e7b19","revision":0,"first_index":1,"fingerprint":"f6-37798f32d2e092fd","name":"shift_x_plus_one of veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":3,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1","2","1"],["2","9","16","14","6","1"],["4","28","85","146","155","104","43","10","1"],["8","76","326","833","1408","1652","1372","806","328","88","14","1"],["16","192","1064","3608","8361","14002","17469","16488","11826","6412","2586","752","149","18","1"],["32","464","3152","13320","39210","85305","142000","184690","189970","155375","101024","51908","20814","6375","1440","226","22","1"],["64","1088","8752","44288","158092","423092","880685","1459810","1956695","2141360","1923493","1420406","860639","425476","169919","53966","13309","2456","319","26","1"],["128","2496","23200","136752","573720","1822548","4553486","9173505","15158272","20786120","23840376","22980412","18658640","12759264","7330180","3520070","1401600","457240","120120","24780","3864","428","30","1"],["256","5632","59392","399616","1926496","7083904","20651344","48978352","96220561","158611106","221417945","263465456","268366964","234573752","176071652","113386736","62483534","29332940","11652910","3880720","1069348","239416","42436","5728","553","34","1"],["512","12544","147968","1118976","6094272","25458144","84827808","231475536","527022162","1014889769","1669752016","2364685622","2898344094","3086262093","2861945376","2313793704","1630761108","1000702146","533291392","245841284","97486876","33001362","9440064","2250648","438906","68181","8112","694","38","1"],["1024","27648","360704","3028480","18387840","86011520","322442400","995016960","2576191380","5675620300","10753517061","17664712562","25314014131","31792739160","35113263215","34180706754","29364534333","22272330024","14906931650","8791709220","4558047886","2070093632","819675726","281193940","82892670","20773548","4363401","753578","104175","11080","851","42","1"],["2048","60416","863744","7972096","53388544","276436864","1151522240","3964679840","11503248680","28534707740","61184918102","114375683009","187641279616","271567916284","348110991316","396412700354","401878394456","363203421944","292814357506","210571931119","134974698496","77004195224","39013720376","17499188924","6920551576","2400461932","725359162","189311023","42203392","7920748","1227380","152834","14696","1024","46","1"],["4096","131072","2037760","20506624","150152960","852568576","3906755072","14846106880","47712422384","131595900480","315021049464","660318036168","1220435354425","2000029880786","2919002365525","3807293302040","4449832172230","4669560608900","4405572102910","3739907732000","2857269904555","1963983964910","1213519141975","673058459120","334396124500","148414157816","58634423212","20527817160","6332860479","1709461710","400183091","80328952","13623830","1914244","216830","19024","1213","50","1"],["8192","282624","4747264","51730432","411121152","2539966208","12697471488","52786416384","186134968032","565045095472","1493717397744","3469837343960","7134140988026","13058529727833","21380213588848","31430153309274","41612170197546","49736457652635","53765569241280","52635750586380","46705927489650","37579520483505","27415945458720","18126918392310","10852145077410","5874561916587","2869982592624","1262280506144","498256657422","175828917667","55206933808","15332280678","3739086494","793446537","144829152","22403004","2879110","299091","24128","1418","54","1"],["16384","606208","10940416","128335872","1100166144","7347207168","39790989056","179631176192","689531870144","2284608168128","6609986561360","16854577026560","38157383382300","77160831169380","140059477112925","229129634462146","338954491024687","454634593118944","554092260845193","614657348187246","621389342528955","572997250002060","482188953879945","370357573501890","259582184434755","165933468248184","96648273470133","51224866719166","24662408132647","10762045739644","4244865591783","1508271558486","480796229277","136818382064","34545517643","7681824410","1490536385","249434900","35463155","4198950","402801","30072","1639","58","1"]],"degrees":[2,5,8,11,14,17,20,23,26,29,32,35,38,41,44],"row_sums":["4","48","576","6912","82944","995328","11943936","143327232","1719926784","20639121408","247669456896","2972033482752","35664401793024","427972821516288","5135673858195456"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1","2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","5","4","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171654,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-ed62d856fc95a05a"]},{"id":171655,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-0a2000766e1dbc24","e-ed62d856fc95a05a"]},{"id":171656,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-0a2000766e1dbc24","e-ed62d856fc95a05a"]},{"id":171657,"position":3,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-0a2000766e1dbc24"]}],"region_exact":{},"incoming_edges":[{"id":"e-0a2000766e1dbc24","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-256337cf537e7b19","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":"77784c68479b9d5633a07902e8762b5a0db6e071076a7b258704f98837727834","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a8bf9bc375c66ce8","role":"sequence","name":"veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]},{"id":"e-ed62d856fc95a05a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-256337cf537e7b19","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":"c4ef057e492be8ac88857f75673713470eae451e7fc39b022064776f65486b96","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-a8bf9bc375c66ce8","role":"sequence","name":"veronese parent of Triangle T(n,k) with diagonals T(n,n-k) = binomial(n,3k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:   1,     2,      1;\n   2:   2,     9,     16,      14,       6,        1;\n   3:   4,    28,     85,     146,     155,      104,       43,        10,         1;\n   4:   8,    76,    326,     833,    1408,     1652,     1372,       806,       328,         88,         14,          1;\n   5:  16,   192,   1064,    3608,    8361,    14002,    17469,     16488,     11826,       6412,       2586,        752,        149,         18,          1;\n   6:  32,   464,   3152,   13320,   39210,    85305,   142000,    184690,    189970,     155375,     101024,      51908,      20814,       6375,       1440,        226,         22,          1;\n   7:  64,  1088,   8752,   44288,  158092,   423092,   880685,   1459810,   1956695,    2141360,    1923493,    1420406,     860639,     425476,     169919,      53966,      13309,       2456,       319,        26,        1;\n   8: 128,  2496,  23200,  136752,  573720,  1822548,  4553486,   9173505,  15158272,   20786120,   23840376,   22980412,   18658640,   12759264,    7330180,    3520070,    1401600,     457240,    120120,     24780,     3864,      428,      30,       1;\n   9: 256,  5632,  59392,  399616, 1926496,  7083904, 20651344,  48978352,  96220561,  158611106,  221417945,  263465456,  268366964,  234573752,  176071652,  113386736,   62483534,   29332940,  11652910,   3880720,  1069348,   239416,   42436,    5728,    553,    34,    1;\n  10: 512, 12544, 147968, 1118976, 6094272, 25458144, 84827808, 231475536, 527022162, 1014889769, 1669752016, 2364685622, 2898344094, 3086262093, 2861945376, 2313793704, 1630761108, 1000702146, 533291392, 245841284, 97486876, 33001362, 9440064, 2250648, 438906, 68181, 8112, 694, 38, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+5*x+4*x^2+x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 2*x + x^2.","mathematica":"P[1, x_] := 1 + 2*x + x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + 5*x + 4*x^2 + x^3)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 2*x + x^2; for(n = 2, N, my(i = n); v[i] = (2 + 5*x + 4*x^2 + x^3)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a8bf9bc375c66ce8","steps":["basis A007318"],"edges":["e-0a2000766e1dbc24"],"text":"Veronese parent of A098172 \u2192 basis A007318"},"headline":"Veronese parent of A098172 \u2192 basis A007318","chain":null}
