{"id":"n-f8e557445427987b","revision":0,"first_index":0,"fingerprint":"f6-5f44c6b5e115ca0a","name":"basis_map_A008277 of Triangle of coefficients of Legendre polynomials 2^n P_n (x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["8","2"],["32","12"],["136","78","6"],["592","460","60"],["2624","2660","540","20"],["11776","15064","4200","280"],["53344","84294","30730","3220","70"],["243392","467436","214452","31080","1260"],["1116928","2575212","1450260","275100","17640","252"],["5149696","14117224","9576600","2282280","203280","5544"],["23835904","77096316","62105604","18101160","2111340","91476","924"],["110690816","419788824","397072104","138687120","20300280","1225224","24024"],["515483648","2280434312","2509955448","1034545512","184744560","14606592","456456","3432"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","10","44","220","1112","5844","31320","171658","957620","5435392","31334624","183342724","1087788392","6540229960"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-4"],["4","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-4"],["0","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2"],["4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196612,"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-66d4f23c7196680e"]},{"id":196613,"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-66d4f23c7196680e"]},{"id":196614,"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-66d4f23c7196680e"]}],"region_exact":{},"incoming_edges":[{"id":"e-66d4f23c7196680e","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-f8e557445427987b","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; 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v[1] = 1; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((4*n - 4 + (4*n - 4)*x)*v[i - 2] + (4*n - 4)*x*deriv(v[i - 2], x) + (4*n - 2)*v[i - 1])/(n)); 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-b2f14d9f44a39df2","steps":["basis A008277"],"edges":["e-66d4f23c7196680e"],"text":"A008556 \u2192 basis A008277"},"headline":"A008556 \u2192 basis A008277","chain":null}