{"id":"n-f4f909be2b0f8048","revision":1,"first_index":1,"fingerprint":"f6-30161f617b2fcf0e","name":"basis_map_A013611 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","8"],["15","64","64"],["105","568","960","512"],["945","5952","13184","12288","4096"],["10395","73032","192640","240640","143360","32768"],["135135","1032576","3088576","4669440","3788800","1572864","262144"],["2027025","16569720","54589248","94750208","94187520","53903360","16515072","2097152"],["34459425","297901440","1060574976","2047467520","2359189504","1669857280","711983104","167772160","16777216"],["654729075","5935802760","22534136064","47386482688","61204340736","50600804352","26886537216","8883535872","1660944384","134217728"],["13749310575","129889690560","520703279424","1175389224960","1664383016960","1552251617280","969423716352","401646551040","105948119040","16106127360","1073741824"],["316234143225","3097457367480","13015292951232","31199578409472","47683923189760","49016851333120","34714758414336","16993260404736","5649979146240","1218025881600","153545080832","8589934592"],["7905853580625","79966307332800","350161982720640","884111803846656","1441694707019776","1606892668846080","1260003771023360","702549577433088","277195561893888","75650480209920","13582834073600","1443109011456","68719476736"],["213458046676875","2222337126630600","10094103992119680","26672314565624832","45998651520307200","54919659715002368","46875243168399360","29048868758880256","13104676790599680","4260127460818944","971940361666560","147626615898112","13400297963520","549755813888"],["6190283353629375","66155441045702400","310507712784515520","854249954340077568","1547339410613907456","1960659343897526272","1798739329603600384","1217419139354722304","612426576998432768","228381110688546816","62267290174881792","12056694754377728","1569621568126976","123145302310912","4398046511104"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","143","2145","36465","692835","14549535","334639305","8365982625","225881530875","6550564395375","203067496256625","6701227376468625","234542958176401875","8678089452526869375"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","8"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168132,"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-1182ea8561b370d4"]},{"id":168133,"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-1182ea8561b370d4"]},{"id":168134,"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-1182ea8561b370d4"]}],"region_exact":{},"incoming_edges":[{"id":"e-1182ea8561b370d4","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-f4f909be2b0f8048","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":"144d9c51d4a831a326b3cf15c3ffad6fcab3d8e37599f9e6bbff33f72ea717a2","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-dba73d4c0da906d3","kind":"oeis_match","node_a_id":"n-f4f909be2b0f8048","node_b_id":null,"fingerprint":null,"oeis_id":"A051583","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":"A051583","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A051583","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-f4f909be2b0f8048","oeis_id":"A051583","target":"row_sums","oeis_offset":0,"first_row":1,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":17,"oeis_terms_available":17,"term_count":200,"raw_size":41415}],"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,          8;\n   3:        15,         64,          64;\n   4:       105,        568,         960,         512;\n   5:       945,       5952,       13184,       12288,        4096;\n   6:     10395,      73032,      192640,      240640,      143360,       32768;\n   7:    135135,    1032576,     3088576,     4669440,     3788800,     1572864,      262144;\n   8:   2027025,   16569720,    54589248,    94750208,    94187520,    53903360,    16515072,    2097152;\n   9:  34459425,  297901440,  1060574976,  2047467520,  2359189504,  1669857280,   711983104,  167772160,   16777216;\n  10: 654729075, 5935802760, 22534136064, 47386482688, 61204340736, 50600804352, 26886537216, 8883535872, 1660944384, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+8*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 + 8*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 + 8*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 A013611"],"edges":["e-1182ea8561b370d4"],"text":"A161198 \u2192 basis A013611"},"headline":"A161198 \u2192 basis A013611","chain":null}