{"id":"n-3c978d7ddc5e48c2","revision":0,"first_index":0,"fingerprint":"f6-7e23844fe6099114","name":"basis_map_A008277 of basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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"],["10","6"],["91","129","27"],["820","2124","1008","108"],["7381","31803","26109","6210","405"],["66430","454530","580608","232362","33210","1458"],["597871","6326865","11888163","7164450","1694196","161595","5103"],["5380840","86660376","231419808","197804376","69070320","10777536","734832","17496"],["48427561","1174941399","4358167065","5089464522","2466709119","560961126","62062686","3175524","59049"],["435848050","15824571534","80236174896","124810787574","80759142450","25128577026","4013984808","331225524","13187610","196830"],["3922632451","212202798333","1453823533647","2958097212594","2488195887102","1017937147437","220212284985","26051421528","1664768457","53045685","649539"],["35303692060","2837328095460","26042176012176","68379609581604","73344637018224","38386550072952","10800344899512","1717116585228","156447963540","7970276556","207852480","2125764"],["317733228541","37863859863555","462607704998181","1551415417939674","2091460955488353","1372763921521392","488529138902832","100196420228892","12195287132886","881906164425","36655434387","796807206","6908733"],["2859599056870","504631146065946","8167138914630816","34702421110922754","58142493704922138","47145544819110738","20794643726274432","5348804020137360","834532668024588","80233817387724","4716552101976","162977011470","2998390122","22320522"],["25736391511831","6719578179137769","143526885600504123","767781744256651554","1584652479449912568","1568883871066227327","844549002649741347","266886224307372672","51878540001230406","6361282891195830","495205934144922","24128072946666","704041345098","11104459695","71744535"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","16","247","4060","71908","1368598","27838243","601865584","13763968051","331553696302","8382161381758","221707694175556","6118269804619057","175728186076357456","5241785394018126343"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["10","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-9","-27","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-27","-18"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-9"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["10","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197365,"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-a0230ed04369df6d"]},{"id":197366,"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-a0230ed04369df6d"]},{"id":197367,"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-a0230ed04369df6d"]}],"region_exact":{},"incoming_edges":[{"id":"e-a0230ed04369df6d","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-3c978d7ddc5e48c2","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":"b9c8e0b296170be57bc4db3f692161a2084582d37910f1353cfeacc206045421","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-08fe62d9c34c34e9","role":"sequence","name":"basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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:        10,           6;\n  2:        91,         129,          27;\n  3:       820,        2124,        1008,          108;\n  4:      7381,       31803,       26109,         6210,         405;\n  5:     66430,      454530,      580608,       232362,       33210,        1458;\n  6:    597871,     6326865,    11888163,      7164450,     1694196,      161595,       5103;\n  7:   5380840,    86660376,   231419808,    197804376,    69070320,    10777536,     734832,     17496;\n  8:  48427561,  1174941399,  4358167065,   5089464522,  2466709119,   560961126,   62062686,   3175524,    59049;\n  9: 435848050, 15824571534, 80236174896, 124810787574, 80759142450, 25128577026, 4013984808, 331225524, 13187610, 196830;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9+27*x+9*x^2)*P(n-2,x) - (27*x+18*x^2)*d/dx P(n-2,x) - 9*x^2*d^2/dx^2 P(n-2,x) + (10+6*x)*P(n-1,x) + 6*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 10 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 10 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-(9 + 27*x + 9*x^2)*P[n - 2, x] - (27*x + 18*x^2)*D[P[n - 2, x], x] - 9*x^2*D[P[n - 2, x], {x, 2}] + (10 + 6*x)*P[n - 1, x] + 6*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] = 10 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -(9 + 27*x + 9*x^2)*v[i - 2] - (27*x + 18*x^2)*deriv(v[i - 2], x) - 9*x^2*deriv(deriv(v[i - 2], x), x) + (10 + 6*x)*v[i - 1] + 6*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-2245b35ec7cd4857","steps":["basis A013610","basis A008277"],"edges":["e-0cc36dd83eba3acf","e-a0230ed04369df6d"],"text":"A091044 \u2192 basis A013610 \u2192 basis A008277"},"headline":"A091044 \u2192 basis A013610 \u2192 basis A008277","chain":null}