{"id":"n-6e91a3cc96dd01b8","revision":0,"first_index":0,"fingerprint":"f6-48fd4cbb64e3687d","name":"basis_map_A013609 of basis_map_A007318 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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","2"],["21","32","12"],["231","562","452","120"],["3465","11664","14648","8128","1680"],["65835","283986","488264","418096","178224","30240"],["1514205","7980048","17477764","20358016","13297264","4616448","665280"],["40883535","254830626","679380876","1004088296","888334544","470372960","137990208","17297280"],["1267389585","9126255456","28705725936","51508163456","57661019744","41231598080","18388885248","4675921920","518918400"],["44358635475","362510186850","1314993093264","2778780402784","3769413248544","3403580604096","2045485318400","788879365632","177143489280","17643225600"],["1729986783525","15823525435200","65060117737596","158342173252608","252600771999008","275977347004416","209109990373248","108494737358848","36886011975936","7419538391040","670442572800"],["74389431691575","753071038621650","3462173130995028","9541238394841176","17512204472566880","22476258345148224","20582778160235136","13447893302106880","6142877484036864","1868252653804032","340449643054080","28158588057600"],["3496303289504025","38816252673030000","197363404933362216","607698168583306560","1261970576373337456","1861945547960043008","2001298457407869696","1578858780569839616","907318333646649088","370380238994485248","101940755298527232","16984137219194880","1295295050649600"],["178311467764705275","2154444050799731250","12006346285252973016","40860776844416745360","94745407824205538256","158057751764629066208","195163498725803504896","180586720679455305216","125216174044471084288","64255308871051202048","23717990469949151232","5963228763105300480","915266908542873600","64764752532480000"],["9807130727058790125","128123242053279303600","776689024432099003380","2895685425846581537664","7417479379929808853520","13809428369561697707264","19269111025209162344512","20471098568563431051264","16638572489136496117504","10295715386309254664192","4774276154883968228352","1608749067348045692928","372354033177544273920","52986474450601574400","3497296636753920000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","65","1365","39585","1464645","65909025","3493178325","213083877825","14702787569925","1132114642884225","96229744645159125","8949366251999798625","903885991451979661125","98523573068265783062625"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-2"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200626,"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-3a46c9d52048c695"]},{"id":200627,"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-3a46c9d52048c695"]},{"id":200628,"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-3a46c9d52048c695"]},{"id":200629,"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-3a46c9d52048c695"]}],"region_exact":{},"incoming_edges":[{"id":"e-3a46c9d52048c695","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-6e91a3cc96dd01b8","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":"169980e2acf326af78e717013c3adc7545ffbea18d9249840e2f16ab97bf4633","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-261c6497576cc5f7","role":"sequence","name":"basis_map_A007318 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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-2b8eeba44cbe04dc","kind":"fingerprint_related","node_a_id":"n-d1461fdc744b2b8a","node_b_id":"n-6e91a3cc96dd01b8","fingerprint":"f6-48fd4cbb64e3687d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-48fd4cbb64e3687d"},"other_id":"n-d1461fdc744b2b8a","other":{"id":"n-d1461fdc744b2b8a","name":"mobius_x_over_x_plus_one of basis_map_A007318 of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","first_index":0,"fingerprint":"f6-48fd4cbb64e3687d","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","3"],["12","32","21"]],"card_rows":[["1"],["2","3"],["12","32","21"],["120","452","562","231"]],"row_sums_preview":["1","5","65","1365","39585","1464645","65909025","3493178325"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4172472abe953cea","steps":["x \u21a6 x+1","x \u21a6 x/(x+1)"],"edges":["e-2a810a02045f5dfd","e-93d1bdabf9a021c2"],"text":"A290319 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)"},"headline":"A290319 \u2192 x \u21a6 x+1 \u2192 x \u21a6 x/(x+1)","chain":null}}],"oeis":[],"bfiles":[],"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,            2;\n  2:          21,           32,            12;\n  3:         231,          562,           452,           120;\n  4:        3465,        11664,         14648,          8128,          1680;\n  5:       65835,       283986,        488264,        418096,        178224,         30240;\n  6:     1514205,      7980048,      17477764,      20358016,      13297264,       4616448,        665280;\n  7:    40883535,    254830626,     679380876,    1004088296,     888334544,     470372960,     137990208,     17297280;\n  8:  1267389585,   9126255456,   28705725936,   51508163456,   57661019744,   41231598080,   18388885248,   4675921920,    518918400;\n  9: 44358635475, 362510186850, 1314993093264, 2778780402784, 3769413248544, 3403580604096, 2045485318400, 788879365632, 177143489280, 17643225600;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n-1+(4*n-2)*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[(4*n - 1 + (4*n - 2)*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] = (4*n - 1 + (4*n - 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2b477f2aa77e3efc","steps":["basis A007318","basis A013609"],"edges":["e-07dfe9939e4be29b","e-3a46c9d52048c695"],"text":"A109692 \u2192 basis A007318 \u2192 basis A013609"},"headline":"A109692 \u2192 basis A007318 \u2192 basis A013609","chain":null}
