{"id":"n-2f0c73de4c291a40","revision":1,"first_index":0,"fingerprint":"f6-e0b26d9c45ca69c1","name":"basis_map_A013611 of Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["3","8"],["12","68","96"],["60","532","1568","1536"],["360","4392","20048","40576","30720"],["2520","39384","245744","765184","1188864","737280"],["20160","385632","3068704","13002304","30936064","39186432","20643840"],["181440","4115808","39958560","215219264","694498304","1342631936","1439760384","660602880"],["1814400","47689920","547754688","3590700800","14692876544","38428258304","62732353536","58437402624","23781703680"],["19958400","597165120","7932898368","61407896320","305249673984","1010425903104","2227186221056","3152105570304","2599094845440","951268147200"],["239500800","8044151040","121470045696","1085942284032","6364943525888","25556096492544","71184974389248","135821460570112","169881783238656","125775390965760","41855798476800"],["3113510400","116070001920","1965229843968","19947811885824","134869495470080","637746543645696","2152097298702336","5182557758095360","8727893289467904","9789405678010368","6581344146554880","2009078326886400"],["43589145600","1786882567680","33548857915392","381461318287872","2925459154643968","15941665375483904","63292182451408896","184464868145856512","391683509473509376","590902130544476160","601187913308307456","370356992197263360","104472072998092800"],["653837184000","29244230668800","603298292520960","7600655817580032","65243721143780352","402950693292320768","1842115997798232064","6311335239466745856","16205285258270605312","30797808488683667456","42108338010115276800","39221878028224167936","22307072658018140160","5850436087893196800"],["10461394944000","507137921740800","11407426520463360","157808390632538112","1499938887355287552","10361834361303953408","53650897562310959104","211508323699361857536","637964678500334436352","1465082051315174998016","2521601917482864476160","3154050329058503294976","2710225844221740318720","1432031336887379558400","351026165273591808000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","176","3696","96096","2978976","107243136","4396968576","202260554496","10315288279296","577656143640576","35237024762075136","2325643634296958976","165120698035084087296","12549173050666390634496"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","4"],["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168076,"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-39cad9e518a532bb"]},{"id":168077,"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-39cad9e518a532bb"]},{"id":168078,"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-39cad9e518a532bb"]},{"id":168079,"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-39cad9e518a532bb"]}],"region_exact":{},"incoming_edges":[{"id":"e-39cad9e518a532bb","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-2f0c73de4c291a40","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":"bf543428a0369c0b00028e91460b054ae4a5266ec0e035fc3c0943f802ae699f","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-2feb6dc01f424b0a","role":"sequence","name":"Triangle read by rows: coefficients of 1; 1(X+2); 1(X+2)(X+3); 1(X+2)(X+3)(X+4); ....","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-a5f7b0099b0a9806","kind":"oeis_match","node_a_id":"n-2f0c73de4c291a40","node_b_id":null,"fingerprint":null,"oeis_id":"A051687","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":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A051687","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A051687","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-2f0c73de4c291a40","oeis_id":"A051687","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":18,"oeis_terms_available":18,"term_count":200,"raw_size":48415}],"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:        3,         8;\n  2:       12,        68,         96;\n  3:       60,       532,       1568,        1536;\n  4:      360,      4392,      20048,       40576,        30720;\n  5:     2520,     39384,     245744,      765184,      1188864,        737280;\n  6:    20160,    385632,    3068704,    13002304,     30936064,      39186432,      20643840;\n  7:   181440,   4115808,   39958560,   215219264,    694498304,    1342631936,    1439760384,     660602880;\n  8:  1814400,  47689920,  547754688,  3590700800,  14692876544,   38428258304,   62732353536,   58437402624,   23781703680;\n  9: 19958400, 597165120, 7932898368, 61407896320, 305249673984, 1010425903104, 2227186221056, 3152105570304, 2599094845440, 951268147200;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+2+(4*n+4)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + 2 + (4*n + 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; for(n = 1, N, my(i = n + 1); v[i] = (n + 2 + (4*n + 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-2feb6dc01f424b0a","steps":["basis A013611"],"edges":["e-39cad9e518a532bb"],"text":"A145324 \u2192 basis A013611"},"headline":"A145324 \u2192 basis A013611","chain":null}
