{"id":"n-2b81ed5ecd810764","revision":0,"first_index":0,"fingerprint":"f6-ac32f0aa28a8fb19","name":"basis_map_A008277 of Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","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"],["9","3"],["44","20"],["265","175","15"],["1854","1554","210"],["14833","15183","3010","105"],["133496","161160","41552","2520"],["1334961","1857231","591885","53550","945"],["14684570","23130470","8777010","1051820","34650"],["176214841","309978823","136508856","20436955","1022175","10395"],["2290792932","4451091996","2231644272","401054940","26966940","540540"],["32071101049","68214438279","38364945619","8051051008","682692010","21126105","135135"],["481066515734","1111703812842","693153586034","166416810560","17060143100","718707990","9459450"],["7697064251745","19203056856735","13146942917550","3554789403165","427869812370","22922349450","472972500","2027025"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","12","64","455","3618","33131","338728","3838572","47678520","644172045","9402091620","147405489205","2470129035710","44053120590540"],"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":{"terms":[{"coeff":{"coeffs":[["1","1"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1"],["0","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"],["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197092,"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-55259740daa24c2e"]},{"id":197093,"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-55259740daa24c2e"]},{"id":197094,"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-55259740daa24c2e"]}],"region_exact":{},"incoming_edges":[{"id":"e-55259740daa24c2e","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-2b81ed5ecd810764","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] = (n + 1 + (n + 1)*x)*v[i - 2] + (n + 1)*x*deriv(v[i - 2], x) + (n + 1)*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-f13c6b611f4ce3bb","steps":["basis A008277"],"edges":["e-55259740daa24c2e"],"text":"A008306 \u2192 basis A008277"},"headline":"A008306 \u2192 basis A008277","chain":null}