{"id":"n-7c95902cf1293962","revision":1,"first_index":0,"fingerprint":"f6-449d2d9631fac89c","name":"basis_map_A013611 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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"],["2","4"],["10","28","16"],["80","264","240","64"],["880","3224","3696","1664","256"],["12320","48656","64640","38080","10240","1024"],["209440","876432","1293504","905920","326400","58368","4096"],["4188800","18366400","29375808","23292416","10151680","2472960","315392","16384"],["96342400","439182400","749109184","653228800","326658304","97484800","17145856","1638400","65536"],["2504902400","11804112000","21233568384","19980385536","11106031104","3841238016","835731456","111181824","8257536","262144"],["72642169600","352338857600","662989931136","664365454080","401996444160","155820026880","39601164288","6567198720","684195840","40632320","1048576"],["2324549427200","11565412121600","22625033226752","23911654255104","15521348029440","6594226636800","1890517364736","368555016192","48163061760","4037017600","196083712","4194304"],["81359229952000","414087621964800","838137811422720","927408031835648","638893798050816","292883324405760","92545014312960","20461495025664","3159927226368","333947863040","23011000320","931135488","16777216"],["3091650738176000","16060766554470400","33505587321922560","38594056455445504","27987596453273600","13685141519622144","4688243841515520","1147716868227072","201923214704640","25329727700992","2210209464320","127427149824","4362076160","67108864"],["126757680265216000","670858031685990400","1437972146416706560","1716378663960955904","1301867680405999616","673041188117602304","246958563580624896","65809366963372032","12869719275798528","1846211694559232","191937498841088","14065351000064","688553721856","20199768064","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","54","648","9720","174960","3674160","88179840","2380855680","71425670400","2357047123200","84853696435200","3309294160972800","138990354760857600","6254565964238592000"],"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":[["-1","4"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168151,"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-4c9f13aa30e5695d"]},{"id":168152,"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-4c9f13aa30e5695d"]},{"id":168153,"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-4c9f13aa30e5695d"]}],"region_exact":{},"incoming_edges":[{"id":"e-4c9f13aa30e5695d","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-7c95902cf1293962","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":"8c701e718405f3ff32ee3badbde0e725b83f48ac68a62cfc7dc4409c9ec2eae3","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-260e022530e5976c","role":"sequence","name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.","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-d661d8b868391290","kind":"oeis_match","node_a_id":"n-7c95902cf1293962","node_b_id":null,"fingerprint":null,"oeis_id":"A034001","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":1,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A034001","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-f44336f1461b4bb6","kind":"oeis_match","node_a_id":"n-7c95902cf1293962","node_b_id":null,"fingerprint":null,"oeis_id":"A052690","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":3,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A052690","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A034001","relation":"match_row_sums"},{"oeis_id":"A052690","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-7c95902cf1293962","oeis_id":"A034001","target":"row_sums","oeis_offset":1,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":17,"oeis_terms_available":17,"term_count":200,"raw_size":43932}],"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:          2,           4;\n  2:         10,          28,          16;\n  3:         80,         264,         240,          64;\n  4:        880,        3224,        3696,        1664,         256;\n  5:      12320,       48656,       64640,       38080,       10240,       1024;\n  6:     209440,      876432,     1293504,      905920,      326400,      58368,      4096;\n  7:    4188800,    18366400,    29375808,    23292416,    10151680,    2472960,    315392,     16384;\n  8:   96342400,   439182400,   749109184,   653228800,   326658304,   97484800,  17145856,   1638400,   65536;\n  9: 2504902400, 11804112000, 21233568384, 19980385536, 11106031104, 3841238016, 835731456, 111181824, 8257536, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-1+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[(3*n - 1 + 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] = (3*n - 1 + 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-260e022530e5976c","steps":["basis A013611"],"edges":["e-4c9f13aa30e5695d"],"text":"A286718 \u2192 basis A013611"},"headline":"A286718 \u2192 basis A013611","chain":null}