{"id":"n-be3f02dc425ece52","revision":1,"first_index":1,"fingerprint":"f6-1840f5afd9477abc","name":"basis_map_A007318 of Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["3","2"],["15","16","4"],["105","142","60","8"],["945","1488","824","192","16"],["10395","18258","12040","3760","560","32"],["135135","258144","193036","72960","14800","1536","64"],["2027025","4142430","3411828","1480472","367920","52640","4032","128"],["34459425","74475360","66285936","31991680","9215584","1630720","173824","10240","256"],["654729075","1483950690","1408383504","740413792","239079456","49414848","6564096","542208","25344","512"],["13749310575","32472422640","32543954964","18365456640","6501496160","1515870720","236675712","24514560","1616640","61440","1024"],["316234143225","774364341870","813455809452","487493412648","186265324960","47868018880","8475282816","1037186304","86211840","4646400","146432","2048"],["7905853580625","19991576833200","21885123920040","13814246935104","5631619949296","1569231121920","307618108160","42880223232","4229668608","288583680","12953600","344064","4096"],["213458046676875","555584281657650","630881499507480","416754915087888","179682232501200","53632480190432","11444151164160","1773002243584","199961498880","16251096576","926914560","35196928","798720","8192"],["6190283353629375","16538860261425600","19406732049032220","13347655536563712","6044294572710576","1914706390524928","439145344141504","74305367392256","9344887954688","871204798464","59382715392","2874540032","93556736","1835008","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","35","315","3465","45045","675675","11486475","218243025","4583103525","105411381075","2635284526875","71152682225625","2063427784543125","63966261320836875"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161931,"position":0,"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-8792ddb513a0c57e"]},{"id":161932,"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-8792ddb513a0c57e"]},{"id":161933,"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-8792ddb513a0c57e"]}],"region_exact":{},"incoming_edges":[{"id":"e-8792ddb513a0c57e","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-be3f02dc425ece52","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":"acbfc6e681c99122b47261877f69d55f4aaafb13366256d1e5128109f314969a","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-7c9c430277778c10","role":"sequence","name":"Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).","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":[{"id":"l-9d63db1e8cc128f3","kind":"oeis_match","node_a_id":"n-be3f02dc425ece52","node_b_id":null,"fingerprint":null,"oeis_id":"A051577","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A051577","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-bc1a545b8684daf9","kind":"oeis_match","node_a_id":"n-be3f02dc425ece52","node_b_id":null,"fingerprint":null,"oeis_id":"A102147","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A102147","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A051577","relation":"match_row_sums"},{"oeis_id":"A102147","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-be3f02dc425ece52","oeis_id":"A051577","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":19,"oeis_terms_available":19,"term_count":200,"raw_size":40593}],"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;\n   2:         3,          2;\n   3:        15,         16,          4;\n   4:       105,        142,         60,         8;\n   5:       945,       1488,        824,       192,        16;\n   6:     10395,      18258,      12040,      3760,       560,       32;\n   7:    135135,     258144,     193036,     72960,     14800,     1536,      64;\n   8:   2027025,    4142430,    3411828,   1480472,    367920,    52640,    4032,    128;\n   9:  34459425,   74475360,   66285936,  31991680,   9215584,  1630720,  173824,  10240,   256;\n  10: 654729075, 1483950690, 1408383504, 740413792, 239079456, 49414848, 6564096, 542208, 25344, 512;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+2*x)*P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(2*n - 1 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 1 + 2*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7c9c430277778c10","steps":["basis A007318"],"edges":["e-8792ddb513a0c57e"],"text":"A161198 \u2192 basis A007318"},"headline":"A161198 \u2192 basis A007318","chain":null}
