{"id":"n-cca66478ca6dde2d","revision":0,"first_index":0,"fingerprint":"f6-e59ec3f9457e398c","name":"basis_map_A008277 of Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2","1"],["5","3"],["17","13","1"],["73","63","7"],["382","358","51","1"],["2365","2347","390","13"],["16937","17527","3241","146","1"],["137861","147275","29445","1610","21"],["1257686","1377466","292255","18315","340","1"],["12714721","14207071","3158160","219035","5115","31"],["141119617","160290399","36993991","2774900","76280","687","1"],["1706150125","1964613651","467609443","37333100","1159970","13545","43"],["22321071242","26001677878","6350195131","533423891","18235910","256487","1253","1"],["314201147513","369658867719","92270173814","8084876317","298435690","4832058","31388","57"]],"degrees":[0,1,1,2,2,3,3,4,4,5,5,6,6,7,7],"row_sums":["1","3","8","31","143","792","5115","37852","316212","2946063","30304133","341255875","4176879877","55224861793","784518364556"],"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","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0"],["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":196807,"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-8bcf7a20a425dc82"]},{"id":196808,"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-8bcf7a20a425dc82"]},{"id":196809,"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-8bcf7a20a425dc82"]}],"region_exact":{},"incoming_edges":[{"id":"e-8bcf7a20a425dc82","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-cca66478ca6dde2d","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 + x; for(n = 2, N, my(i = n + 1); v[i] = (1 + x)*v[i - 2] + x*deriv(v[i - 2], x) + n*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-bf8bba6616804a77","steps":["basis A008277"],"edges":["e-8bcf7a20a425dc82"],"text":"A249247 \u2192 basis A008277"},"headline":"A249247 \u2192 basis A008277","chain":null}