{"id":"n-2a6cccf5866a7948","revision":0,"first_index":0,"fingerprint":"f6-6231079da586bb57","name":"basis_map_A008277 of basis_map_A013611 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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":false,"rows":[["1"],["11","8"],["76","132","32"],["521","1748","1008","128"],["3571","21180","22080","6464","512"],["24476","245516","414912","211008","37632","2048"],["167761","2777400","7183664","5662336","1709056","205824","8192"],["1149851","30976928","118449648","136108032","61920000","12391424","1077248","32768"],["7881196","342568212","1893468768","3054188864","1966883072","582042624","82952192","5455872","131072"],["54018521","3768798188","29653854720","65467355456","57393146112","23570291712","4898611200","522731520","26935296","524288"],["370248451","41330285940","457989544784","1359440983680","1579944899328","865388418048","246055161856","37884592128","3141271552","130285568","2097152"],["2537720636","452345340956","7005510460800","27595869513600","41719103473152","29661696592896","11084365078528","2309658640384","274021023744","18166841344","619708416","8388608"],["17393796001","4944619347120","106434048731040","550984121617920","1068342913037568","966876833482752","462016048644096","125137628135424","19922313216000","1877127987200","101791563776","2906652672","33554432"],["119218851371","54008042661368","1609283697893280","10866838636151040","26734104382862592","30349384459591680","18178920476819456","6225571520675840","1277246706941952","160387589472256","12292095737856","555397152768","13472104448","134217728"],["817138163596","589620725333412","24248853590313632","212356084756824960","657283412437406208","925402921106942976","684482056571125760","290485547502796800","74694985340223488","12003776047087616","1219177066004480","77490731089920","2962520801280","61807263744","536870912"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","19","240","3405","53807","935592","17714233","362105899","7935571872","185356267013","4591677788487","120123902783064","3306657779766001","95468725831133635","2882847767878248784"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["11","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197260,"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-714264db8ac8bca4"]},{"id":197261,"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-714264db8ac8bca4"]},{"id":197262,"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-714264db8ac8bca4"]}],"region_exact":{},"incoming_edges":[{"id":"e-714264db8ac8bca4","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-2a6cccf5866a7948","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"e74c272c56506f36b6af5b8efc138d5e45d9016a852ec76bcf11ed5f02fdacf0","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-b2e2c5159c22bcc9","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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":[],"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:       11,          8;\n  2:       76,        132,          32;\n  3:      521,       1748,        1008,         128;\n  4:     3571,      21180,       22080,        6464,         512;\n  5:    24476,     245516,      414912,      211008,       37632,        2048;\n  6:   167761,    2777400,     7183664,     5662336,     1709056,      205824,       8192;\n  7:  1149851,   30976928,   118449648,   136108032,    61920000,    12391424,    1077248,     32768;\n  8:  7881196,  342568212,  1893468768,  3054188864,  1966883072,   582042624,   82952192,   5455872,   131072;\n  9: 54018521, 3768798188, 29653854720, 65467355456, 57393146112, 23570291712, 4898611200, 522731520, 26935296, 524288;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 11 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 11 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 11 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); 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-353fb5a029e4e669","steps":["basis A013611","basis A008277"],"edges":["e-a3f64a2b3fc7c9ce","e-714264db8ac8bca4"],"text":"A207607 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A207607 \u2192 basis A013611 \u2192 basis A008277","chain":null}