{"id":"n-00bc3b0d8b47bf75","revision":1,"first_index":0,"fingerprint":"f6-6f5c8093b03ddefa","name":"basis_map_A013610 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"],["2","3"],["8","30","27"],["48","300","612","405"],["384","3408","11196","16092","8505"],["3840","44448","203976","463212","519534","229635"],["46080","660096","3914496","12289752","21520404","19900242","7577955"],["645120","11038464","80546688","324721872","780585984","1117899144","882200808","295540245"],["10321920","205645824","1785477888","8820150912","27101859984","53012755584","64420674408","44427680280","13299311025"],["185794560","4228042752","42626539008","249822088704","937661176224","2336424459696","3863222674128","4085152639848","2505199292730","678264862275"],["3715891200","95151144960","1093529217024","7426154497536","32993082580608","100175176238688","210440647685232","301906745222256","282957686325936","156361656931110","38661097149675"],["81749606400","2327426334720","30052164907008","232267739618304","1193695550118144","4282418079829440","10940730352112448","19899709199059248","25245194048172720","21266290691018388","10701328523952780","2435649120429525"],["1961990553600","61498954874880","881844374863872","7648025129422848","44675167236498432","185143026874058496","558064375958930112","1232503415073180864","1978964591891233392","2252309365908358992","1724205942255135492","796847247043050420","168059789309637225"],["51011754393600","1746122118266880","27540375362076672","264986981479784448","1735156232855672832","8164356241462903296","28395400790486570112","73899916988822460864","143890835519660632992","206982387905459838192","213752556941760447192","150033474092254472820","64133098050279349350","12604484198222791875"],["1428329123020800","53023371417354240","912566401717764096","9650405885762174976","70048320019821379584","369149629622270791680","1456384077692119130112","4369225139716441083264","10014836670645117053760","17450664538445386741728","22750645014711539414928","21514894386865721461512","13948438146880434080220","5547706499622865469850","1020963220056046141875"]],"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_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-3"],["2","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167415,"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-ad9ddfe0012b7a14"]},{"id":167416,"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-ad9ddfe0012b7a14"]},{"id":167417,"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-ad9ddfe0012b7a14"]},{"id":167418,"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-ad9ddfe0012b7a14"]}],"region_exact":{},"incoming_edges":[{"id":"e-ad9ddfe0012b7a14","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-00bc3b0d8b47bf75","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":"64cce4d75bb84f7f0f72289212c0095094a1186c63b69b7942454df818c2f195","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-2b477f2aa77e3efc","role":"sequence","name":"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-997f32f34bc9d210","kind":"oeis_match","node_a_id":"n-00bc3b0d8b47bf75","node_b_id":null,"fingerprint":null,"oeis_id":"A147625","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":"A147625","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A147625","relation":"match_row_sums"}],"bfiles":[{"node_id":"n-00bc3b0d8b47bf75","oeis_id":"A147625","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":51869}],"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,          3;\n  2:         8,         30,          27;\n  3:        48,        300,         612,          405;\n  4:       384,       3408,       11196,        16092,         8505;\n  5:      3840,      44448,      203976,       463212,       519534,        229635;\n  6:     46080,     660096,     3914496,     12289752,     21520404,      19900242,       7577955;\n  7:    645120,   11038464,    80546688,    324721872,    780585984,    1117899144,     882200808,     295540245;\n  8:  10321920,  205645824,  1785477888,   8820150912,  27101859984,   53012755584,   64420674408,   44427680280,   13299311025;\n  9: 185794560, 4228042752, 42626539008, 249822088704, 937661176224, 2336424459696, 3863222674128, 4085152639848, 2505199292730, 678264862275;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n+(6*n-3)*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[(2*n + (6*n - 3)*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] = (2*n + (6*n - 3)*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 A013610"],"edges":["e-ad9ddfe0012b7a14"],"text":"A109692 \u2192 basis A013610"},"headline":"A109692 \u2192 basis A013610","chain":null}