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T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["3","24"],["10","144","243"],["35","720","2430","1728"],["126","3360","17010","25920","10125"],["462","15120","102060","241920","212625","52488"],["1716","66528","561330","1814400","2551500","1469664","250047"],["6435","288288","2918916","11975040","23388750","22044960","9001692","1119744"],["24310","1235520","14594580","72648576","182432250","242494560","165031020","50388480","4782969"],["92378","5250960","70887960","415134720","1277025750","2206700496","2145403260","1108546560","263063295","19683000"],["352716","22170720","336717810","2268414720","8270262000","17653603968","22526734230","16812956160","6839645670","1299078000","78594219"],["1352078","93117024","1571349780","11972188800","50507671500","128619114624","204242390352","201755473920","119693799225","39405366000","6130349082","306110016"],["5200300","389398464","7228208988","61457235840","294628083750","872772563520","1663116607152","2057905833984","1627835669460","788107320000","214562217870","27856011456","1167575877"],["20058300","1622493600","32855495400","308403583488","1656464559750","5600290615920","12473374553640","18619148021760","18557326631844","12058041996000","4863410271720","1114240458240","122595467085","4374822312"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","27","397","4913","56541","624675","6715185","70743825","733632265","7511788379","76108530213","764298282401","7617031926661","75412169029059"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-1","3","0"],["5","-21","0"],["-8","54","-9"],["4","-54","27"],["0","18","-27"],["0","0","9"]]},"terms":[{"coeff":{"coeffs":[["2","18","54","54"],["-14","-102","-243","-135"],["36","180","198","0"],["-40","-60","306","243"],["16","-108","-495","-162"],["0","72","144","-81"],["0","0","36","81"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","9","27","27"],["0","-6","-42","-90","-54"],["0","12","48","27","-27"],["0","-8","12","144","108"],["0","0","-36","-117","-27"],["0","0","0","-18","-54"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181388,"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-347182bc7f49467b"]},{"id":181389,"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-347182bc7f49467b"]},{"id":181390,"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-347182bc7f49467b"]},{"id":181391,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-347182bc7f49467b"]}],"region_exact":{},"incoming_edges":[{"id":"e-347182bc7f49467b","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-832c931ce4d0668a","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"29ea5d48439ee69788b4d759cd5a3131f76a9a27e61b6229c42331da3a7f9882","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-bf08de87964eed6e","role":"sequence","name":"basis_map_A013610 of Triangle read by rows. 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